2.2 Kinematics in One Dimension | General Physics

Chad's Prep
6 Sept 202329:30
EducationalLearning
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TLDRIn this educational video, Chad explores kinematics in one dimension, focusing on problem-solving techniques for scenarios with constant velocity and uniform acceleration. He emphasizes understanding the concepts' relationships and provides equations for calculating displacement, velocity, and acceleration. Through examples, Chad demonstrates how to apply these principles, clarifying that the average velocity and time factors play crucial roles in determining the overall motion and distance traveled.

Takeaways
  • πŸ“š Kinematics in one dimension is the focus of the lesson, aiming to develop problem-solving skills and an intuitive understanding of displacement, velocity, and acceleration.
  • πŸ”’ There are two primary scenarios in one-dimensional kinematics: constant velocity (no acceleration) and uniform acceleration (constant acceleration).
  • πŸš— For constant velocity, the equation Ξ”x = v * Ξ”t applies, where Ξ”x is displacement, v is velocity, and Ξ”t is the change in time.
  • πŸƒβ€β™‚οΈ In the case of uniform acceleration, the average velocity is used, and the displacement is calculated using the equation Ξ”x = (v_initial + v_final) / 2 * t.
  • πŸ“ˆ The concept of acceleration is key, defined as the change in velocity over the change in time, and can be represented as a positive or negative value depending on the direction of change.
  • πŸ”„ When dealing with uniform acceleration, it's important to consider the square terms, as they are commonly forgotten and can lead to errors in calculations.
  • πŸ“Š The lesson emphasizes the importance of systematically working through the available equations to solve kinematics problems, even if the initial conditions are not immediately clear.
  • 🧠 Developing an intuitive feel for the relationships between displacement, velocity, and acceleration is crucial for demystifying kinematics problems.
  • πŸ“ For a round-trip journey with different average speeds for each leg, the overall average speed is influenced by the time spent at each speed, and it may not be the simple average of the two speeds.
  • πŸ›£οΈ In problems involving travel over a distance with varying speeds, it's essential to calculate the time spent at each speed to determine the overall average speed correctly.
  • πŸ“Š The lesson provides examples and practical applications of kinematics equations, highlighting the importance of understanding the underlying concepts and being able to apply them to real-world scenarios.
Q & A
  • What is the main topic of the lesson?

    -The main topic of the lesson is kinematics in one dimension, focusing on problem-solving in the area of kinematics with an emphasis on understanding displacement, velocity, and acceleration.

  • What are the two scenarios that will be dealt with in this lesson?

    -The two scenarios are constant velocity (no acceleration) and uniform acceleration (constant acceleration).

  • What is the single equation used for kinematics calculations when there is no acceleration?

    -The single equation used when there is no acceleration is Delta X (displacement) equals velocity times time (Ξ”X = V Γ— Ξ”T).

  • How is the average velocity calculated for uniform acceleration?

    -The average velocity for uniform acceleration is calculated as the average of the initial and final velocities (V_avg = (V_initial + V_final) / 2).

  • What is the significance of the equation Ξ”X = V_avgT for uniform acceleration?

    -The equation Ξ”X = V_avgT is significant for uniform acceleration because it allows us to calculate the displacement when the velocity is changing at a constant rate over time.

  • Why might students struggle with kinematics problems involving varying acceleration?

    -Students might struggle with varying acceleration because it requires more complex calculations, potentially involving calculus, which can be challenging for those taking an algebra-based physics class.

  • What is the role of the equation Ξ”X = V_initialT + (1/2)aT^2 in the context of uniform acceleration?

    -This equation accounts for the change in velocity due to constant acceleration, where 'a' is the acceleration and 'T' is the time. It helps calculate the displacement by considering the initial velocity, the time during which the object is moving, and the effect of acceleration on the displacement.

  • How does the instructor approach the concept of acceleration in the lesson?

    -The instructor approaches the concept of acceleration by explaining it as the change in velocity per second and emphasizes understanding its intuitive meaning to solve problems more effectively.

  • What is the average speed of a round trip if a person travels 60 miles to work at an average speed of 40 miles per hour and 60 miles back home at an average speed of 60 miles per hour?

    -The average speed of the round trip is not simply the average of the two speeds (40 mph and 60 mph), but rather it is closer to 40 mph due to the longer time spent traveling at the lower speed.

  • How does the instructor demonstrate the concept of uniform acceleration over time?

    -The instructor demonstrates the concept by using a car accelerating from rest with an acceleration of 10 meters per second squared, and showing how the velocity increases by 10 m/s every second, leading to increasing displacement with each successive second.

  • What is the displacement of the car in the first second of its uniform acceleration from rest?

    -The displacement in the first second is calculated by using the average velocity for that interval, which is half of the initial velocity (0 m/s) and the final velocity (10 m/s), resulting in an average velocity of 5 m/s and a displacement of 5 meters.

Outlines
00:00
πŸ“š Introduction to Kinematics in One Dimension

The paragraph introduces the topic of kinematics in one dimension, emphasizing the challenges students may face in problem-solving within this area. The speaker, Chad, outlines his goal to provide an intuitive understanding of the material and a systematic approach to problem-solving. He welcomes viewers to his educational platform, which offers comprehensive resources for various science subjects. Chad reviews the concepts of displacement, velocity, and acceleration from the previous lesson and hints at the focus on constant velocity and uniform acceleration scenarios, clarifying that varying acceleration will not be covered in this algebra-based physics class.

05:00
πŸš€ Understanding Uniform Acceleration and Displacement

This paragraph delves into the specifics of uniform acceleration, explaining how it differs from constant velocity. Chad discusses the use of average velocity in calculations when dealing with uniform acceleration and highlights the importance of understanding the relationship between initial and final velocities. He critiques the common practice of presenting a fifth equation in textbooks, arguing that it adds unnecessary confusion for students. Chad's approach is to simplify the concept by focusing on the four main equations relevant to uniform acceleration, making the subject more accessible and less intimidating.

10:04
🧠 Developing Intuition for Acceleration and Time

Chad continues to build on the concept of uniform acceleration by discussing the least favorite equation involving squares, which he finds difficult for mental calculations. He emphasizes the importance of recognizing that not all equations are necessary for solving problems, especially when an intuitive understanding of the concepts can be applied. Chad then introduces a problem-solving scenario involving a car's displacement and average speed, illustrating how to apply the concepts of velocity and time to find the solution.

15:08
πŸ›£οΈ Calculating Displacement and Average Speed in Real-World Scenarios

The paragraph presents a real-world scenario of a man commuting to work and back home, highlighting the misconception that the average speed of a round trip is the simple average of the individual average speeds. Chad explains why this is incorrect and provides a detailed calculation to find the true average speed for the entire round trip. He emphasizes the importance of considering the time spent at each average speed and how it affects the overall average velocity.

20:09
🏎️ Detailed Analysis of Uniform Acceleration Over Time

Chad tackles a multi-part question involving a car accelerating uniformly from rest. He explains how to calculate the velocity after a certain time and the displacement during each successive second. Chad uses both intuitive reasoning and formal equations to demonstrate how the car's displacement increases with each second due to the constant acceleration. He emphasizes the importance of understanding acceleration and its impact on velocity and displacement over time.

25:15
πŸ“ Multiple Methods for Calculating Displacement During Uniform Acceleration

In this paragraph, Chad explores different methods for calculating the displacement of an object undergoing uniform acceleration. He uses the example of an object moving from rest andεŠ ι€Ÿ uniformly, discussing various equations that can be applied to find the displacement during different time intervals. Chad highlights the flexibility in choosing the most straightforward method for problem-solving, depending on the given information and the learner's preference. He reinforces the concept by using different approaches to arrive at the same result, showcasing the interconnectedness of the equations and their applications.

Mindmap
Determining velocity and displacement during acceleration from rest.
Calculating average speed of a round trip.
Car's displacement with constant speed.
Use corresponding equations based on the type of acceleration.
Identify if the scenario involves no acceleration or uniform acceleration.
Introduction of average velocity for varying speeds.
Displacement formula derived from velocity's definition.
Specific equations for scenarios with no acceleration and uniform acceleration.
Limited number of equations simplifies problem-solving.
Acceleration: Zero (constant velocity) or uniform (constant acceleration).
Varying acceleration only relevant for calculus-based classes.
Previous lesson introduced calculations involving constant velocity and uniform acceleration.
Understanding the relationship between velocity, time, and acceleration for intuitive problem solving.
Use of average velocity for problems involving variable speeds.
Approaching problems by dissecting given quantities and identifying applicable equations.
Emphasis on understanding acceleration's impact on velocity and displacement.
Practical Examples
Systematic Approach
Equations of Kinematics
Displacement, Velocity, and Acceleration
Courses offered in Chemistry and Physics, targeting various exams.
Focus on intuitive understanding and systematic problem-solving.
Chad's Prep: Aim to ease the learning of science.
Insights & Strategies
Problem Solving in Kinematics
Core Concepts
Introduction to Kinematics
Kinematics in One Dimension
Alert
Keywords
πŸ’‘Kinematics
Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. In the video, the topic of kinematics, specifically in one dimension, is the focus of the lesson, aiming to help students understand and solve problems related to the motion of objects along a straight line.
πŸ’‘Displacement
Displacement is a vector quantity that represents the change in position of an object. In the context of the video, displacement is calculated as the product of velocity and time. It is used to determine the total distance an object has moved from its initial position to its final position, and is a key concept in solving one-dimensional kinematics problems.
πŸ’‘Velocity
Velocity is a physical quantity that describes the rate of change of an object's position with respect to time, and it is a vector quantity that includes both magnitude (speed) and direction. In the video, the definition of velocity is used to derive equations for calculating displacement, and understanding velocity is crucial for solving problems involving constant speed as well as uniform acceleration.
πŸ’‘Acceleration
Acceleration is the rate of change of velocity of an object with respect to time, and it is typically measured in meters per second squared (m/s^2). In the video, the concept of uniform acceleration, where an object's speed changes at a constant rate, is discussed. This is a fundamental concept for solving kinematics problems involving changing speeds.
πŸ’‘Uniform Acceleration
Uniform acceleration refers to a type of motion where an object's acceleration remains constant throughout the motion. In the video, the instructor explains how to calculate displacement and velocity for objects undergoing uniform acceleration, emphasizing the use of specific kinematic equations that account for this constant rate of change in speed.
πŸ’‘Problem Solving
Problem solving in the context of the video involves applying kinematic principles and equations to determine unknown quantities such as displacement, velocity, and acceleration. The instructor aims to develop an intuitive understanding and a systematic approach for students to tackle kinematics problems, both with constant velocity and uniform acceleration scenarios.
πŸ’‘Average Velocity
Average velocity is defined as the mean speed of an object over a certain distance or time interval. In the video, the concept of average velocity is used to derive equations for calculating displacement when an object is undergoing uniform acceleration. It is particularly useful when the velocity is changing and cannot be simply equated with the initial or final velocity alone.
πŸ’‘Initial and Final Velocities
Initial and final velocities are the velocities of an object at the beginning and end of a specified time interval or displacement. In the video, these velocities are crucial for calculating the average velocity and for using in kinematic equations to find other unknowns, such as displacement and acceleration, especially in scenarios with uniform acceleration.
πŸ’‘Calculus-based Physics
Calculus-based physics is a more advanced approach to physics that incorporates calculus to deal with problems involving rates of change, such as acceleration and varying acceleration. In the video, it is mentioned that while calculus-based physics might cover varying acceleration, the algebra-based class being discussed will focus on constant velocity and uniform acceleration scenarios.
πŸ’‘Master Course
The term 'Master Course' refers to comprehensive, in-depth educational programs designed to thoroughly cover a subject area. In the video, the instructor mentions that for those joining his master course, the problems discussed are already typed out in study guides, suggesting a structured learning path with additional resources for deeper understanding of the material.
Highlights

The lesson focuses on kinematics in one dimension, specifically problem-solving in the area of kinematics with an emphasis on developing an intuitive feel for the material.

Two main scenarios are discussed: constant velocity (no acceleration) and uniform acceleration (constant acceleration).

For constant velocity, the equation Ξ”x = velocity Γ— time is used, derived from the definition of velocity (Ξ”x/Ξ”T).

In the case of uniform acceleration, the average velocity is used in the equation Ξ”x = (initial velocity + final velocity) / 2 Γ— time.

The lesson aims to demystify kinematics problems and provide a systematic foundation for approaching them.

Displacement, velocity, and acceleration are the key quantities analyzed in the lesson, with their interrelationships being crucial for solving problems.

For uniform acceleration, the equation Ξ”x = initial velocity Γ— time + 0.5 Γ— acceleration Γ— timeΒ² is introduced, accounting for the changing velocity.

The lesson clarifies that in algebra-based physics classes, varying acceleration is not typically dealt with, unlike in calculus-based physics classes.

The concept of average velocity is crucial for solving problems involving uniform acceleration, where the speed is not constant and changes at a constant rate.

The lesson provides a method for determining the total displacement when given constant speed and time, using the equation Ξ”x = velocity Γ— time.

A man's round trip average speed is discussed, highlighting that it's not simply the average of the two speeds involved, but depends on the time spent at each speed.

The lesson explains how to calculate the average speed for the entire round trip by considering the time spent at each average speed.

A multi-part question involving a car accelerating uniformly from rest is used to illustrate the concepts of velocity and displacement over time intervals.

The velocity after a certain time period is calculated by adding the product of acceleration and time to the initial velocity.

The lesson demonstrates that the displacement in each successive second increases due to the constant acceleration, using both intuitive reasoning and equations.

Different methods for calculating displacement in each second are presented, including using the average velocity and the formula Ξ”x = initial velocity Γ— time + 0.5 Γ— acceleration Γ— timeΒ².

The lesson concludes by showing that various equations can be used to solve for displacement, but some methods are more intuitive or easier depending on the scenario.

Transcripts
00:00

kinematics in one dimension going to be

00:03

the topic of this lesson we're really

00:04

going to do some hardcore problem

00:06

solving in the area of kinematics

00:08

specifically in one dimension and this

00:10

is where some students begin to have a

00:11

little bit of trouble but my goal for

00:13

this lesson is that you really get a

00:15

good intuitive feel for much of the

00:18

material as well as a systematic

00:20

approach for problem solving my name is

00:22

Chad and welcome to Chad's prep where my

00:25

goal is to take the stress out of

00:26

learning science now if you're new to

00:28

the channel we've got comprehensive

00:29

playlists for General chemistry organic

00:31

chemistry General Physics and high

00:33

school chemistry and on chatsprep.com

00:35

you'll find premium Master courses for

00:36

the same that include study guides and a

00:38

ton of practice you'll also find

00:40

comprehensive prep courses for the DAT

00:42

the MCAT and the oat

00:45

now in the last lesson on displacement

00:46

velocity and acceleration I hinted

00:48

toward the fact that we might do some

00:50

calculations involving two different

00:52

scenarios and the first is constant

00:54

velocity where you have no acceleration

00:56

and the second was going to be when you

00:58

have uniform acceleration or constant

01:00

acceleration and then I said but you're

01:02

not going to be doing any problems where

01:04

you have varying acceleration well

01:05

there's one caveat to that if you're

01:07

taking a calculus based physics class

01:08

you might you still might not but you

01:11

might but for an algebra based physical

01:12

class like the one we're gonna you know

01:14

the one we're going through here you're

01:16

not going to be dealing with a varying

01:17

acceleration so in the two situations we

01:20

will deal with so again no acceleration

01:22

again you're moving at constant velocity

01:24

or with a uniform acceleration a

01:26

constant acceleration these are the two

01:27

situations and it turns out there's only

01:29

a handful of equations for which you're

01:31

going to do kinematics calculations and

01:33

what's nice is when you don't have an

01:34

acceleration there's one equation that's

01:35

all you got so we'll see that this

01:37

equation is related to the definition of

01:39

velocity so but even with uniform

01:41

acceleration constant acceleration

01:42

you've only got a handful of equations

01:45

but it's enough to where some students

01:47

get to the point where they just are

01:49

given a bunch of quantities in a problem

01:51

and they just don't even know where to

01:52

start well the good news again is that

01:54

there's a limited number of equations

01:55

and even if you just kind of

01:56

systematically work your way through all

01:58

the equations because you recognize it

01:59

was uniform acceleration you'd

02:00

eventually figure some things out

02:03

but hopefully we kind of demystify this

02:05

and we're going to give you kind of a

02:06

systematic foundation for kind of

02:08

approaching this as well as a little bit

02:09

more of an intuitive understanding of

02:11

what displacement velocity and

02:12

acceleration are and their relationships

02:14

to each other

02:16

all right so before we dive into some

02:17

calculations and some problems that

02:19

we're going to solve throughout this

02:20

lesson I just want to kind of look at

02:21

these lovely quantities and kind of see

02:23

where most of them come from so if we

02:25

take a look at the first one here so

02:27

Delta X displacement equals velocity

02:29

times time and again this really is just

02:31

coming from uh the definition of

02:33

velocity if you recall velocity

02:36

was equal to Delta X over delta T

02:39

displacement over time and if you

02:40

rearrange this you can get Delta x

02:42

equals V times delta T so in fact some

02:46

people actually write this and I see

02:48

some people some textbooks write this as

02:49

Delta x equals V delta T so if I'm going

02:52

to leave it as T it's pretty common to

02:53

see it this way and the idea is that if

02:55

delta T is T final minus t initial where

02:57

T initial is zero then all you've got is

02:59

T final and if that's the if that's the

03:01

only time point you have then we can

03:02

just call it t we don't have to actually

03:03

signify it something else and so that's

03:05

where this kind of reduces down to this

03:07

form right here so uh when you've got no

03:09

acceleration you're at constant velocity

03:11

this is the only equation you got

03:13

nothing else to worry about

03:15

all right so if we take a look at

03:17

uniform acceleration so now your speed

03:20

is not constant it is changing but it's

03:22

changing at the same rate the entire

03:24

time and you've got a very similar

03:26

looking equation where you can't just

03:28

use the velocity because it's changing

03:30

so it's it's you know it's either always

03:31

going up or always going down it's not

03:33

at one set point but what you can do is

03:35

you can use a directly analogous

03:36

equation in which you use the average

03:38

velocity and if you're calling the last

03:40

lesson we actually came up with an

03:42

equation for the average velocity where

03:43

we say that the average velocity was

03:45

equal to V initial plus v final over two

03:49

or one-half times the sum of V initial

03:51

plus v final or something like that also

03:53

one thing to note I said you might see V

03:55

initial and you might see V naught and

03:57

all my acquired equations are written

03:59

with a not so I'm going to try and use

04:01

that here but again that essentially

04:03

means velocity at time zero so when you

04:05

see that little not symbol it means at

04:07

Time Zero

04:08

all right so you know if we can

04:10

calculate the average velocity because

04:12

we know both the initial and the final

04:13

velocities then we could use this

04:15

equation and we vary analogous to what

04:18

we're doing with no acceleration

04:20

now this next one's a little bit

04:21

different here and so it turns out it's

04:23

not going to be easy for me to show you

04:25

where it's derived from at least not the

04:27

second part of it because it actually

04:29

does come from calculus it involves

04:30

taking a second derivative which we're

04:33

just not going to do in this algebra

04:34

based class but we could

04:36

so the idea is this so you know if you

04:39

ignore that last term this looks pretty

04:41

much a lot like both of these so but the

04:44

key is that you know if you just use

04:45

your initial velocity

04:47

well you know the idea is that that

04:49

velocity is changing the whole time it's

04:50

either going up or going down and how do

04:52

you account for that well that's what

04:54

this last term does it accounts for it

04:55

and so if you're speeding up so then

04:58

along with that initial velocity times

05:00

time you have to add an additional term

05:02

to get an overall higher displacement

05:04

because your velocity been going up so

05:06

you're going to go further and further

05:07

and further and definitely further than

05:08

if you just maintained that initial

05:10

velocity the entire time on the other

05:12

hand if you're slowing down well if

05:15

you're slowing down that would actually

05:17

mean you have a negative acceleration

05:18

we'll get into this a little bit so but

05:20

if you're slowing down that means your

05:21

acceleration and your velocity point in

05:23

opposite directions and that needs you

05:24

may need to make one positive and one

05:26

negative and it's customary at this

05:27

point to make the velocity positive and

05:28

the acceleration negative and so in this

05:31

case you'd be plugging in a negative

05:32

number for a right there and so you'd

05:35

end up with this first term and then

05:36

you'd be subtracting or adding a

05:38

negative term overall now one thing I

05:41

just want to bring this up and give you

05:43

one caveat is that a lot of textbooks

05:45

will take this and give you a fifth

05:47

equation and they'll have V not T minus

05:50

one half

05:51

a t squared and they'll say well always

05:54

plug in a positive value for a you just

05:57

have to choose the right equation if

05:59

you're speeding up it's this equation if

06:00

you're slowing down this equation if

06:02

you're accelerating at this equation if

06:03

you're decelerating it's this equation

06:04

so I hate that I really do so we're

06:07

going to make a point of making you know

06:10

of making terms positive negative a lot

06:12

of times and stuff like that throughout

06:13

this course especially in kinematics and

06:15

as a result I just find no need to

06:18

include this equation giving students an

06:19

extra equation often adds to the

06:21

confusion making them realize there's

06:23

only four uniform acceleration equations

06:25

usually puts a smile on students faces

06:27

so I'm not going to use this equation so

06:30

what I'm going to do is is input an

06:33

acceleration that's either positive or

06:35

negative right there and again if you're

06:36

speeding up then the acceleration of the

06:38

Velocity are in the same direction I'll

06:39

make it positive if you're slowing down

06:41

decelerating then the acceleration of

06:43

velocity point in opposite directions

06:44

and I'll make the acceleration negative

06:46

in that case

06:49

all right they're also not going to show

06:50

where this one's dry from this is my

06:52

least favorite of the equations so it's

06:54

my least favorite because it has squares

06:55

in it and that makes it a little

06:57

difficult sometimes to do the math in

06:59

your head and I like doing math in my

07:01

head when I can but definitely we'll

07:03

plug and chug with a calculator when

07:04

appropriate and things of A Sort uh but

07:06

one thing that's nice about this

07:07

equation right here is this is the only

07:09

one of all the equations here that does

07:12

not have time in it notice this one has

07:13

time in it this one has time in it this

07:15

one has time in it this is the only one

07:17

that does not have time in it and so if

07:21

you're doing a calculation involving

07:22

uniform acceleration but you don't know

07:24

anything about the time that might be

07:26

the indication that this is the lovely

07:27

equation you're using

07:29

and then finally this last one here this

07:31

last one just like these first couple

07:33

were a rearrangement of the definition

07:35

of velocity this last one is really just

07:37

a rearrangement of the definition of

07:39

acceleration so if you recall

07:41

acceleration was equal to the change in

07:44

velocity

07:45

over the change in time and so what you

07:48

might see is that acceleration equals V

07:50

final minus V initial all over and again

07:53

it's really T final minus t initial but

07:55

if T initial is zero then it's really

07:56

just T final but if there's only a final

07:58

if there's only one time point we'll

07:59

just make it t

08:01

so at that point so it's like T minus

08:03

zero

08:04

and then rearranging it we get a t

08:07

equals V final minus V initial or V

08:11

final equals V initial plus a t adding

08:15

it to the other side and notice that

08:17

looks just like that in Reverse so

08:20

that's just really a rearrangement of

08:22

the definite definition of acceleration

08:24

and in fact if the numbers are nice

08:27

I would recommend that if you get an

08:29

intuitive feel you might not even need

08:31

this equation because you understand

08:33

what acceleration actually means and

08:36

we'll demonstrate this throughout this

08:37

lesson

08:38

all right so now we've seen these

08:39

equations again the first thing you're

08:40

going to do when approaching any

08:41

kinematics problem so is first you're

08:44

going to ask yourself a question is no

08:46

acceleration I.E constant velocity or is

08:48

it under uniform acceleration and the

08:50

problem is going to have to tell you and

08:52

clue you in in one way shape or form and

08:53

usually it's going to be by directly

08:55

telling you which of those two scenarios

08:57

you're in let's do some plug it in chug

09:00

it all right once again if you're

09:01

joining me for my master course all

09:03

these problems are already typed out for

09:05

you on the study guides and there's

09:07

plenty of room so you can work them out

09:08

right on there if you so choose so but

09:11

for the rest of you I will make sure

09:12

that the questions are put up on the

09:13

screen as we go here

09:15

all right so first question here a car

09:17

travels due north for four hours with a

09:19

constant speed of 60 miles per hour what

09:22

is the displacement again for those of

09:24

us that are in the states that's the

09:25

majority of my audience I'm using units

09:28

you're going to be somewhat more

09:28

familiar with to start but definitely be

09:31

transitioning over to SI units in a

09:33

little bit so in this case travels due

09:35

north for four hours with a constant

09:37

speed the moment you hear constant speed

09:39

you think constant velocity no

09:41

acceleration that's the only equation we

09:43

got

09:44

all right we want to know what is the

09:46

total displacement well if this is the

09:47

only equation we got and it's an

09:49

equation that involves displacement and

09:50

that's the one we're gonna have to use

09:51

and so in this case we've just got the

09:53

displacement equals the velocity times

09:56

the time and we'll just do some plugging

09:57

and chugging here velocity was given as

09:59

60 miles per hour and the time was given

10:04

four hours I didn't make a sure it was

10:05

in hours that the units would cancel

10:07

here

10:09

so and 60 times four

10:15

is 240 miles so uh and there's a good

10:20

chance you probably did this one in your

10:22

head without like plugging and chugging

10:24

going through an equation like Well Chad

10:26

the car goes 60 miles per hour that

10:29

means every hour it goes 60 miles so if

10:32

you did that for one one hour that would

10:33

be 60 miles if you did it for two hours

10:35

that'd be 120 miles three hours 180

10:37

miles four hours

10:39

240 miles and if you did it that way and

10:42

that's the level of an intuitive

10:43

understanding you have fantastic so if

10:45

you didn't hopefully that last exercise

10:47

of one two three four hours how far it

10:49

went uh was instructive but the answer

10:52

definitely is the displacement is 240

10:54

miles all right second problem we'll

10:56

take a look at this one's going to be a

10:57

little bit tricky but it says a man

11:00

travels 60 miles to work one way at an

11:02

average speed of 40 miles per hour he

11:04

travels 60 miles home at an average

11:06

speed of 60 miles per hour what is the

11:08

average speed of his round trip now you

11:11

might be tend to be like well 40 miles

11:12

per hour one way 60 miles per hour the

11:14

other way the average is 50 right Chad

11:17

wrong it turns out but we'll see why it

11:19

is a little bit tricky this makes a

11:21

great multiple choice question on a test

11:23

because they can put that 50 miles per

11:24

hour is definitely one of the answer

11:25

choices and you might be tempted to just

11:27

pick it without even trying to work the

11:28

problem out but it's not going to work

11:30

out to be 50 as we'll see let's see if

11:32

we can map this out a little bit so

11:33

we've got home

11:38

and over here we've got work

11:42

and so on his way to work

11:45

so 60 miles to work so that is a

11:47

displacement or distance technically but

11:49

displacement

11:51

of 60 miles and we're told is average

11:54

velocity is 40 miles

11:58

per hour

11:59

all right then on the way home he's just

12:01

retracing his steps it's 60 miles back

12:03

home so again that's a displacement of

12:06

60 miles but now maybe he's avoiding

12:09

rush hour or something his average

12:11

velocity is now 60 miles

12:15

per hour

12:17

okay so again

12:19

your intuitive your intuition if it's

12:22

telling you the average is going to be

12:23

50 again it kind of looks that way

12:24

you're like well it's 40 miles an hour

12:26

on the way there 60 miles an hour on the

12:28

way back the average 50. except the

12:29

problem is he doesn't spend the same

12:31

amount of time going those two average

12:32

speeds so if you look and you say does

12:34

he get to work faster or does he get

12:36

home faster well he's you know

12:39

traveling faster on the way home so it

12:41

should take him less time to get home

12:42

and that's the key here he's going to

12:44

spend less time averaging 60 miles an

12:46

hour than he does averaging 40 miles an

12:49

hour and because he spends more time at

12:51

an average of 40 miles per hour the

12:54

overall average for the entire round

12:55

trip is going to come out closer to 40

12:57

than it is to 60. so 50 is right in the

13:00

middle it's going to be lower than 50

13:01

miles per hour that's what your

13:03

intuition hopefully in the future will

13:05

start to tell you

13:07

all right so we got to work this out

13:08

here uh in this case whether we've got

13:10

acceleration here or not we're not

13:12

really given any indication so but we're

13:14

given average velocities and given

13:16

average velocities that's fantastic we

13:17

can go straight here and that way

13:19

whether he starts and stops and speeds

13:21

up and slows down all the way to work it

13:22

doesn't matter we've got the only thing

13:24

we need is that average velocity we can

13:26

use Delta x equals V average t

13:28

and for all we know maybe he just went

13:30

constantly 40 miles per hour out of the

13:32

way work and never stop there's no

13:33

lights no nothing whatever the key is

13:36

once we know an average velocity though

13:37

it doesn't matter and we can just go

13:39

straight to displacement equals average

13:41

velocity times time

13:43

all right so what we want to do here is

13:45

again get the average velocity for his

13:47

entire round-trip commute and so let's

13:50

put that up on the board here so the

13:51

round

13:53

trip

13:55

so and for the round trip Delta x equals

13:57

V

13:58

average T so if you rearrange that that

14:01

average velocity is going to equal

14:04

Delta X over t

14:07

where we need to know the displacement

14:08

for the entire round trip and the time

14:10

for the entire round trip well one of

14:12

those is not so bad to figure out if

14:14

it's 60 miles to work and 60 miles home

14:16

well then this entire displacement for

14:18

the entire round trip I was tired

14:20

distance really we should say is 120

14:22

miles this displacement would be zero

14:24

that's tricky so 120 miles for the

14:26

distance notice really dealing with

14:28

distance and speed scalars rather than

14:30

velocity and displacement here

14:33

all right so 120 miles is a total

14:35

distance traveled but the time we don't

14:36

have that so we've got to find the time

14:39

for on the way to work and the time for

14:40

on the way home and add them together to

14:42

get there and to do that we're actually

14:44

going to use the same set of equations

14:45

here so on the way to work we've got

14:49

Delta x equals V average T if you

14:52

rearrange and solve for T you're going

14:54

to get Delta X let's see if I can write

14:57

that correctly over V average

15:00

which in this case was 60

15:02

miles over 40 miles per hour

15:08

is going to come out to 60 over 40 the

15:09

same thing as 6 over 4 which reduces

15:12

down to 3 over 2 or 1.5

15:17

hours

15:19

all right going home this one you've

15:20

probably seen your head if he's

15:22

traveling it's an average of 60 miles

15:24

per hour then how long does it take him

15:26

to go 60 miles well one hour but again

15:30

you could set it up just the same way

15:31

and just say time is equal to the

15:34

distance over the speed the average

15:36

speed which is 60 miles over 60

15:41

miles per hour which is going to get you

15:44

one

15:46

hour and then you're simply going to add

15:48

these together in one and a half hours

15:49

plus one hour is two and a half

15:53

hours

15:55

cool and that's we're going to plug in

15:57

time for the entire time of the round

15:59

trip

16:05

and so here we've got 120 miles over two

16:07

and a half hours and you can definitely

16:09

plug this in your calculator you could

16:11

also if you wanted to and I like doing

16:12

math in my head when I can instead of

16:14

two and a half I can write this as five

16:16

halves hours so and dividing by five

16:19

halves is the same thing as multiplying

16:21

by two-fifths so five goes into 120 24

16:24

times times two

16:27

it's going to get you

16:29

48 miles

16:33

per hour that is the answer to this

16:36

question

16:39

cool and as we pointed out before

16:41

because he spends more time at 40 miles

16:43

per hour than he does at 60 miles per

16:45

hour the overall average velocity for

16:48

the entire round trip should come out

16:49

closer to 40 than it does to 60 and

16:52

indeed that's true all right the next

16:54

question here is a multi-part question

16:55

it says a car accelerates uniformly from

16:58

rust with an acceleration of 10 meters

17:00

per second squared what is the velocity

17:02

after six seconds how far does it travel

17:04

in the first second how far does it

17:06

travel in the second second how far does

17:08

it travel in the third second and the

17:10

idea is that if this car is speeding up

17:12

then it's going to be going faster and

17:13

faster and faster and so how far it

17:15

travels every additional second should

17:17

be further and further and further so

17:20

let's see kind of how we approach this

17:21

here so we're told that the uniform

17:23

acceleration that acceleration is 10 and

17:25

notice it centimeters per second squared

17:27

but you might have intuited that I'm

17:29

going to write meters per second per

17:30

second to make that pretty so it means

17:33

it's speeding up 10 meters per second

17:35

every second nice round number it's

17:38

going to make doing this not so bad at

17:40

so first question we want is what is the

17:43

velocity

17:44

after six seconds so and then we want

17:46

how far does it travel in the first the

17:48

second and the third second so velocity

17:50

after six seconds this you can do in

17:52

your head so the acceleration is 10

17:55

meters per second per second so we're

17:58

told it accelerates from rest which

18:00

means the initial velocity is zero it's

18:02

a big keyword from rest means your

18:04

initial velocity is zero so if you start

18:06

at zero and you're speeding up 10 meters

18:08

per second every second then one second

18:11

after you start you should be moving

18:13

with a velocity of 10 meters per second

18:15

after two seconds you'll there's an

18:18

additional 10 meters per second for that

18:19

second second now you're up to 20 meters

18:22

per second for the velocity after three

18:24

seconds up to 30 meters per second for

18:26

the velocity after four seconds up to 40

18:28

meters per second for the velocity again

18:30

the key is that the acceleration is 10

18:32

meters per second every second per

18:35

second so every second should go further

18:38

the velocity goes up by 10 meters per

18:39

second that's how this works and so

18:41

you're like well then what is the

18:42

velocity after six seconds well if you

18:44

started 0 it should be 60 meters per

18:46

second and hopefully you can kind of see

18:48

that but if you understand what

18:50

acceleration means that it's just the

18:51

change in the velocity per second and

18:54

it's telling you that it's changing by

18:55

10 meters per second every second then

18:57

after six seconds it'd be 60 meters per

19:00

second after 10 seconds would be 100

19:02

meters per second after 15 seconds it'd

19:05

be 150 meters per second and what you

19:07

may not have realized if you're doing it

19:09

in your head like that is that you're

19:11

technically just using that equation

19:12

without actually having to think about

19:14

using that equation and if you can think

19:15

of acceleration like that that's exactly

19:18

where I want you to get and that's why

19:19

the problems we're going to do are going

19:20

to be nice round numbers because if

19:21

instead I'd said instead of saying the

19:23

acceleration is 10 meters per second

19:24

squared what if I'd said the

19:26

acceleration is 1.436 meters per second

19:28

squared well now all of a sudden the

19:30

math is tough to do in your head you're

19:32

going to put out your calculator so and

19:34

all of a sudden it's not going to be

19:35

this kind of intuitive thing that you're

19:37

probably doing in your head but if we

19:39

start with nice round numbers then when

19:41

you do get harder numbers hopefully

19:43

you've got some of those intuitive

19:44

Pathways built in and you might still

19:46

realize you need to pull out your

19:48

calculator but you might recognize what

19:49

you need to do in your calculator

19:50

without even thinking about it as an

19:52

actual equation

19:54

so but if we did use that equation

19:56

we'd say that the final velocity equals

19:58

the initial velocity of zero plus the

20:01

acceleration of 10 meters per second

20:05

per second times the time

20:08

of six seconds and you can see yep that

20:11

final velocity after six seconds

20:13

is going to be 60 meters per second

20:16

notice these seconds cancel and you'll

20:18

look for units of meters per second okay

20:20

so same thing we just intuitively

20:22

calculated now we use an equation to do

20:24

it as well

20:25

now let's go to the other problems here

20:26

and so the question is how far does it

20:28

travel in the first second the second

20:29

second and so we want Delta X

20:33

for zero two one second or maybe we'll

20:37

write this a little bit different

20:42

so T equals zero

20:46

to t equal one second and then we want

20:48

the displacement for T equals one second

20:52

to T equals two seconds so the second

20:55

second and then we want the displacement

20:59

for T equals two seconds to T equals

21:03

three seconds

21:05

cool now there's actually not just one

21:07

way of approaching this there's a few

21:10

different ways a couple at least a

21:11

couple of different ways and we'll go

21:13

through a couple of them but we want

21:15

displacement so you got to say okay well

21:16

which of these equations have

21:17

displacement well

21:19

the first three all have displacement

21:22

that means you're probably using one of

21:24

those first three probably not using

21:25

that last one so at least not directly

21:29

or initially as we'll see let's kind of

21:32

take an accounting here and figure out

21:34

how we might do this well in this case

21:36

first equation Delta x equals V average

21:38

T let's write that out

21:45

well for that first second how long a

21:47

period of time is passing well one

21:49

second okay we got the time do we have

21:51

the average velocity during that for a

21:53

second well we don't but you might

21:54

recall again that your average velocity

21:58

is equal to V initial plus v final over

22:01

two well for that first second at Time

22:05

Zero what was the velocity well he

22:06

started from rest the velocity was Zero

22:08

and after one second we haven't figured

22:10

that out yet formally but we did

22:13

informally we said the acceleration is

22:16

10 meters per second per second so after

22:18

one second we said yeah the velocity at

22:20

that point would be 10 meters per second

22:22

after two seconds it would be 20 meters

22:24

per second after three seconds it would

22:26

be 30 meters per second and so even

22:27

though we haven't formally done it we

22:29

can calculate it or just reason it out

22:31

in our heads based on knowing what

22:32

acceleration is or use this equation to

22:35

figure it out and say that V final

22:37

equals the initial velocity of zero plus

22:39

acceleration of 10 meters per second per

22:41

second times one second and you're going

22:43

to get a velocity of 10 meters per

22:44

second so even though we're not given

22:48

what that final velocity is at T equals

22:50

one second we can Intuit it or calculate

22:53

it very easily and so yeah we don't know

22:55

the average velocity off the topper

22:57

heads but we can get it pretty quickly

22:58

and totally use this so the initial

23:00

velocity was Zero the final velocity at

23:02

T equals one second is 10 meters per

23:04

second so the average is going to come

23:05

up to 5 meters per second

23:07

and from there

23:12

and again from zero to one is one second

23:14

and so five meters per second times one

23:16

second

23:19

gonna get us five meters

23:21

so from T equals zero to T equals one

23:23

second the displacement is five meters

23:26

so what about for the second second from

23:29

T equals one second to T equals two

23:30

seconds how far now well again this

23:32

whatever this object is it is speeding

23:34

up and going faster and faster and

23:36

faster which means every successive

23:38

second we should anticipate that it's

23:39

going further and further and further

23:41

now we don't know the answer yet but

23:43

it's got to be higher than five meters

23:44

right

23:45

let's figure this out we could do this

23:47

the same way and we could rationalize

23:49

that on our head so at T equals one

23:51

second we already figured out that at T

23:53

equals one second

23:54

the velocity is 10

23:57

meters per second

23:58

at T equals two seconds we're going to

24:01

be up to

24:02

a velocity of 20 meters per second and

24:07

if you know the initial and final

24:08

velocities then the average is halfway

24:09

in between or you can do 10 plus 20 over

24:12

2. so and the average would come out to

24:14

15 meters per second and we could do the

24:17

same thing and say the displacement

24:19

equals the average velocity 15 meters

24:21

per second and again from T equals one

24:23

it's equals two as a total duration of

24:25

time of one second so 15 meter second

24:27

times one second is 15 meters and we

24:31

haven't written anything on the board

24:32

yet we just kind of rationalized that

24:33

out and did it in our head using again

24:35

the exact same equation okay now that's

24:39

not the only way to do it but we could

24:40

have done it that way and it's 15 meters

24:42

it's longer than 5 meters just like we

24:43

expected it to be so but let's approach

24:46

this a little bit different way let's

24:48

see if we can go about this route right

24:50

here

24:51

and so in this case

24:54

Delta x equals V initial t plus one half

24:58

a

24:59

t squared okay

25:01

so in this case v initial well V initial

25:04

at T equals one second is 10 meters per

25:07

second

25:08

I'm sorry yes 10 meters per second

25:14

how long is the entire duration of this

25:16

journey from T equals one second to

25:17

equals took it total of one second

25:23

what's our acceleration

25:25

10 meters per second per second and

25:27

again how long is the entire duration of

25:29

this part of the journey again it's a

25:31

full one second long

25:34

Square it

25:35

so by the way big mistake students make

25:37

is forgetting the square terms not just

25:40

here but another place like equals mc

25:41

squared that's another common place

25:42

where students forget to actually Square

25:44

it they might write it in the equation

25:45

then forget to do it in the math so this

25:47

is a key common place where that happens

25:48

here in physics

25:50

so let's look at this out so notice 10

25:52

meters per second times one second

25:53

that's 10 meters

25:54

and then 10 meters per second per second

25:56

times one second squared is also going

25:58

to come out to 10 meters then times a

26:00

half is 5 meters 10 meters plus 5 meters

26:02

gets us one other way of getting us our

26:07

15 meters

26:11

so more than one way to skin a cat so to

26:14

speak

26:15

all right last one here from two seconds

26:17

to three seconds

26:18

well from two seconds to three seconds

26:20

we can see again that at T equals two

26:22

seconds our velocity

26:24

equals 20 meters per second by the time

26:27

we get to T equals three seconds it'll

26:29

have gone up an additional 10 meters per

26:31

second based on the acceleration and

26:33

should be up to 30.

26:36

meters per second

26:38

which means the average velocity during

26:39

that one second duration somewhere

26:41

exactly right halfway in between of 25

26:43

meters per second

26:45

so and then the whole time is one second

26:49

so you say Okay average velocity is 25

26:51

meters per second times one second and

26:53

it should be 25 meters and again we just

26:56

did it in our head or we could use this

26:58

equation again and say okay the initial

27:00

velocity is 20 meters per second times

27:03

again the whole duration from two

27:04

seconds to three seconds is one second

27:07

so we got 20 meters per second times one

27:09

second plus one half times 10 meters per

27:11

second per second times one second

27:13

squared and you're going to get 20 plus

27:15

5 and get 25 meters per second yet again

27:17

but maybe there's another way we can do

27:20

this

27:21

because displacement also shows up in

27:23

this equation right here now again this

27:25

is my least favorite equation I would

27:26

never use this one for this calculation

27:28

I just want to demonstrate that it will

27:30

lead us to the same answer

27:32

okay so in this case we've got

27:36

V final squared equals V initial squared

27:39

plus two a Delta X so and this one is

27:43

kind of the most laborious because again

27:46

we've already figured it out that the

27:47

time at T equals two seconds the initial

27:49

velocity for this period is 20 meter

27:51

second the final is 30 meter second

27:53

but we had to figure those out and

27:55

they're both going to show up here that

27:56

we would have to figure those out in

27:58

this one as well so alien will get a

28:00

square in which makes the math big and

28:01

stuff and that's not very fun but that

28:03

final velocity was 30 meters per second

28:05

and so here we plug that in 30 meters

28:08

per second and square it and in fact I'm

28:10

going to subtract off this initial

28:13

velocity squared as well and it's 20

28:14

meters per second

28:18

and then we'll square it and that's

28:20

going to equal our two times our

28:22

acceleration 10 meters per second per

28:24

second

28:26

foreign

28:27

times Delta X here well 30 squared is

28:30

920 squared is 400 900 minus 400 is 500.

28:36

and then 500 divided by 20 which I might

28:40

do differently I might do 500 divided by

28:41

the 10. so is 50 and then divided by the

28:45

2 is 25 and yet again I get Delta x

28:48

equals 25 meters get the same answer

28:53

anyway here now it won't always work out

28:55

like this like oh you just pick any

28:56

other questions it worked out in this

28:57

one that we could have used any one of

28:58

those three equations

28:59

but which one's the easiest and

29:01

personally for me

29:03

the easiest one for me was again

29:05

that guy right there I could pretty

29:07

quickly knowing what acceleration

29:09

actually means figure out that the

29:10

initial velocity at T equals two seconds

29:12

so it's 20 meters per second and at T

29:14

equals three seconds was 30 meters per

29:15

second and at the average is going to be

29:17

halfway in between at 25 meters per

29:18

second and so during that one second

29:20

period 25 meters per second times a

29:22

second is 25 meters that was definitely

29:24

the easiest way for me but all three of

29:27

those equations could have worked

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