2.2 Kinematics in One Dimension | General Physics
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
- π 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.
- π’ There are two primary scenarios in one-dimensional kinematics: constant velocity (no acceleration) and uniform acceleration (constant acceleration).
- π For constant velocity, the equation Ξx = v * Ξt applies, where Ξx is displacement, v is velocity, and Ξt is the change in time.
- πββοΈ In the case of uniform acceleration, the average velocity is used, and the displacement is calculated using the equation Ξx = (v_initial + v_final) / 2 * t.
- π 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.
- π 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.
- π 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.
- π§ Developing an intuitive feel for the relationships between displacement, velocity, and acceleration is crucial for demystifying kinematics problems.
- π 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.
- π£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.
- π 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 (ΞX = V Γ ΞT).
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 ΞX = V_avgT for uniform acceleration?
-The equation ΞX = 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 ΞX = 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
π 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.
π 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.
π§ 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.
π£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.
ποΈ 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.
π 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 andε ι 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
Keywords
π‘Kinematics
π‘Displacement
π‘Velocity
π‘Acceleration
π‘Uniform Acceleration
π‘Problem Solving
π‘Average Velocity
π‘Initial and Final Velocities
π‘Calculus-based Physics
π‘Master Course
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 Ξx = velocity Γ time is used, derived from the definition of velocity (Ξx/ΞT).
In the case of uniform acceleration, the average velocity is used in the equation Ξx = (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 Ξx = 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 Ξx = 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 Ξx = 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
kinematics in one dimension going to be
the topic of this lesson we're really
going to do some hardcore problem
solving in the area of kinematics
specifically in one dimension and this
is where some students begin to have a
little bit of trouble but my goal for
this lesson is that you really get a
good intuitive feel for much of the
material as well as a systematic
approach for problem solving my name is
Chad and welcome to Chad's prep where my
goal is to take the stress out of
learning science now if you're new to
the channel we've got comprehensive
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you'll find premium Master courses for
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comprehensive prep courses for the DAT
the MCAT and the oat
now in the last lesson on displacement
velocity and acceleration I hinted
toward the fact that we might do some
calculations involving two different
scenarios and the first is constant
velocity where you have no acceleration
and the second was going to be when you
have uniform acceleration or constant
acceleration and then I said but you're
not going to be doing any problems where
you have varying acceleration well
there's one caveat to that if you're
taking a calculus based physics class
you might you still might not but you
might but for an algebra based physical
class like the one we're gonna you know
the one we're going through here you're
not going to be dealing with a varying
acceleration so in the two situations we
will deal with so again no acceleration
again you're moving at constant velocity
or with a uniform acceleration a
constant acceleration these are the two
situations and it turns out there's only
a handful of equations for which you're
going to do kinematics calculations and
what's nice is when you don't have an
acceleration there's one equation that's
all you got so we'll see that this
equation is related to the definition of
velocity so but even with uniform
acceleration constant acceleration
you've only got a handful of equations
but it's enough to where some students
get to the point where they just are
given a bunch of quantities in a problem
and they just don't even know where to
start well the good news again is that
there's a limited number of equations
and even if you just kind of
systematically work your way through all
the equations because you recognize it
was uniform acceleration you'd
eventually figure some things out
but hopefully we kind of demystify this
and we're going to give you kind of a
systematic foundation for kind of
approaching this as well as a little bit
more of an intuitive understanding of
what displacement velocity and
acceleration are and their relationships
to each other
all right so before we dive into some
calculations and some problems that
we're going to solve throughout this
lesson I just want to kind of look at
these lovely quantities and kind of see
where most of them come from so if we
take a look at the first one here so
Delta X displacement equals velocity
times time and again this really is just
coming from uh the definition of
velocity if you recall velocity
was equal to Delta X over delta T
displacement over time and if you
rearrange this you can get Delta x
equals V times delta T so in fact some
people actually write this and I see
some people some textbooks write this as
Delta x equals V delta T so if I'm going
to leave it as T it's pretty common to
see it this way and the idea is that if
delta T is T final minus t initial where
T initial is zero then all you've got is
T final and if that's the if that's the
only time point you have then we can
just call it t we don't have to actually
signify it something else and so that's
where this kind of reduces down to this
form right here so uh when you've got no
acceleration you're at constant velocity
this is the only equation you got
nothing else to worry about
all right so if we take a look at
uniform acceleration so now your speed
is not constant it is changing but it's
changing at the same rate the entire
time and you've got a very similar
looking equation where you can't just
use the velocity because it's changing
so it's it's you know it's either always
going up or always going down it's not
at one set point but what you can do is
you can use a directly analogous
equation in which you use the average
velocity and if you're calling the last
lesson we actually came up with an
equation for the average velocity where
we say that the average velocity was
equal to V initial plus v final over two
or one-half times the sum of V initial
plus v final or something like that also
one thing to note I said you might see V
initial and you might see V naught and
all my acquired equations are written
with a not so I'm going to try and use
that here but again that essentially
means velocity at time zero so when you
see that little not symbol it means at
Time Zero
all right so you know if we can
calculate the average velocity because
we know both the initial and the final
velocities then we could use this
equation and we vary analogous to what
we're doing with no acceleration
now this next one's a little bit
different here and so it turns out it's
not going to be easy for me to show you
where it's derived from at least not the
second part of it because it actually
does come from calculus it involves
taking a second derivative which we're
just not going to do in this algebra
based class but we could
so the idea is this so you know if you
ignore that last term this looks pretty
much a lot like both of these so but the
key is that you know if you just use
your initial velocity
well you know the idea is that that
velocity is changing the whole time it's
either going up or going down and how do
you account for that well that's what
this last term does it accounts for it
and so if you're speeding up so then
along with that initial velocity times
time you have to add an additional term
to get an overall higher displacement
because your velocity been going up so
you're going to go further and further
and further and definitely further than
if you just maintained that initial
velocity the entire time on the other
hand if you're slowing down well if
you're slowing down that would actually
mean you have a negative acceleration
we'll get into this a little bit so but
if you're slowing down that means your
acceleration and your velocity point in
opposite directions and that needs you
may need to make one positive and one
negative and it's customary at this
point to make the velocity positive and
the acceleration negative and so in this
case you'd be plugging in a negative
number for a right there and so you'd
end up with this first term and then
you'd be subtracting or adding a
negative term overall now one thing I
just want to bring this up and give you
one caveat is that a lot of textbooks
will take this and give you a fifth
equation and they'll have V not T minus
one half
a t squared and they'll say well always
plug in a positive value for a you just
have to choose the right equation if
you're speeding up it's this equation if
you're slowing down this equation if
you're accelerating at this equation if
you're decelerating it's this equation
so I hate that I really do so we're
going to make a point of making you know
of making terms positive negative a lot
of times and stuff like that throughout
this course especially in kinematics and
as a result I just find no need to
include this equation giving students an
extra equation often adds to the
confusion making them realize there's
only four uniform acceleration equations
usually puts a smile on students faces
so I'm not going to use this equation so
what I'm going to do is is input an
acceleration that's either positive or
negative right there and again if you're
speeding up then the acceleration of the
Velocity are in the same direction I'll
make it positive if you're slowing down
decelerating then the acceleration of
velocity point in opposite directions
and I'll make the acceleration negative
in that case
all right they're also not going to show
where this one's dry from this is my
least favorite of the equations so it's
my least favorite because it has squares
in it and that makes it a little
difficult sometimes to do the math in
your head and I like doing math in my
head when I can but definitely we'll
plug and chug with a calculator when
appropriate and things of A Sort uh but
one thing that's nice about this
equation right here is this is the only
one of all the equations here that does
not have time in it notice this one has
time in it this one has time in it this
one has time in it this is the only one
that does not have time in it and so if
you're doing a calculation involving
uniform acceleration but you don't know
anything about the time that might be
the indication that this is the lovely
equation you're using
and then finally this last one here this
last one just like these first couple
were a rearrangement of the definition
of velocity this last one is really just
a rearrangement of the definition of
acceleration so if you recall
acceleration was equal to the change in
velocity
over the change in time and so what you
might see is that acceleration equals V
final minus V initial all over and again
it's really T final minus t initial but
if T initial is zero then it's really
just T final but if there's only a final
if there's only one time point we'll
just make it t
so at that point so it's like T minus
zero
and then rearranging it we get a t
equals V final minus V initial or V
final equals V initial plus a t adding
it to the other side and notice that
looks just like that in Reverse so
that's just really a rearrangement of
the definite definition of acceleration
and in fact if the numbers are nice
I would recommend that if you get an
intuitive feel you might not even need
this equation because you understand
what acceleration actually means and
we'll demonstrate this throughout this
lesson
all right so now we've seen these
equations again the first thing you're
going to do when approaching any
kinematics problem so is first you're
going to ask yourself a question is no
acceleration I.E constant velocity or is
it under uniform acceleration and the
problem is going to have to tell you and
clue you in in one way shape or form and
usually it's going to be by directly
telling you which of those two scenarios
you're in let's do some plug it in chug
it all right once again if you're
joining me for my master course all
these problems are already typed out for
you on the study guides and there's
plenty of room so you can work them out
right on there if you so choose so but
for the rest of you I will make sure
that the questions are put up on the
screen as we go here
all right so first question here a car
travels due north for four hours with a
constant speed of 60 miles per hour what
is the displacement again for those of
us that are in the states that's the
majority of my audience I'm using units
you're going to be somewhat more
familiar with to start but definitely be
transitioning over to SI units in a
little bit so in this case travels due
north for four hours with a constant
speed the moment you hear constant speed
you think constant velocity no
acceleration that's the only equation we
got
all right we want to know what is the
total displacement well if this is the
only equation we got and it's an
equation that involves displacement and
that's the one we're gonna have to use
and so in this case we've just got the
displacement equals the velocity times
the time and we'll just do some plugging
and chugging here velocity was given as
60 miles per hour and the time was given
four hours I didn't make a sure it was
in hours that the units would cancel
here
so and 60 times four
is 240 miles so uh and there's a good
chance you probably did this one in your
head without like plugging and chugging
going through an equation like Well Chad
the car goes 60 miles per hour that
means every hour it goes 60 miles so if
you did that for one one hour that would
be 60 miles if you did it for two hours
that'd be 120 miles three hours 180
miles four hours
240 miles and if you did it that way and
that's the level of an intuitive
understanding you have fantastic so if
you didn't hopefully that last exercise
of one two three four hours how far it
went uh was instructive but the answer
definitely is the displacement is 240
miles all right second problem we'll
take a look at this one's going to be a
little bit tricky but it says a man
travels 60 miles to work one way at an
average speed of 40 miles per hour he
travels 60 miles home at an average
speed of 60 miles per hour what is the
average speed of his round trip now you
might be tend to be like well 40 miles
per hour one way 60 miles per hour the
other way the average is 50 right Chad
wrong it turns out but we'll see why it
is a little bit tricky this makes a
great multiple choice question on a test
because they can put that 50 miles per
hour is definitely one of the answer
choices and you might be tempted to just
pick it without even trying to work the
problem out but it's not going to work
out to be 50 as we'll see let's see if
we can map this out a little bit so
we've got home
and over here we've got work
and so on his way to work
so 60 miles to work so that is a
displacement or distance technically but
displacement
of 60 miles and we're told is average
velocity is 40 miles
per hour
all right then on the way home he's just
retracing his steps it's 60 miles back
home so again that's a displacement of
60 miles but now maybe he's avoiding
rush hour or something his average
velocity is now 60 miles
per hour
okay so again
your intuitive your intuition if it's
telling you the average is going to be
50 again it kind of looks that way
you're like well it's 40 miles an hour
on the way there 60 miles an hour on the
way back the average 50. except the
problem is he doesn't spend the same
amount of time going those two average
speeds so if you look and you say does
he get to work faster or does he get
home faster well he's you know
traveling faster on the way home so it
should take him less time to get home
and that's the key here he's going to
spend less time averaging 60 miles an
hour than he does averaging 40 miles an
hour and because he spends more time at
an average of 40 miles per hour the
overall average for the entire round
trip is going to come out closer to 40
than it is to 60. so 50 is right in the
middle it's going to be lower than 50
miles per hour that's what your
intuition hopefully in the future will
start to tell you
all right so we got to work this out
here uh in this case whether we've got
acceleration here or not we're not
really given any indication so but we're
given average velocities and given
average velocities that's fantastic we
can go straight here and that way
whether he starts and stops and speeds
up and slows down all the way to work it
doesn't matter we've got the only thing
we need is that average velocity we can
use Delta x equals V average t
and for all we know maybe he just went
constantly 40 miles per hour out of the
way work and never stop there's no
lights no nothing whatever the key is
once we know an average velocity though
it doesn't matter and we can just go
straight to displacement equals average
velocity times time
all right so what we want to do here is
again get the average velocity for his
entire round-trip commute and so let's
put that up on the board here so the
round
trip
so and for the round trip Delta x equals
V
average T so if you rearrange that that
average velocity is going to equal
Delta X over t
where we need to know the displacement
for the entire round trip and the time
for the entire round trip well one of
those is not so bad to figure out if
it's 60 miles to work and 60 miles home
well then this entire displacement for
the entire round trip I was tired
distance really we should say is 120
miles this displacement would be zero
that's tricky so 120 miles for the
distance notice really dealing with
distance and speed scalars rather than
velocity and displacement here
all right so 120 miles is a total
distance traveled but the time we don't
have that so we've got to find the time
for on the way to work and the time for
on the way home and add them together to
get there and to do that we're actually
going to use the same set of equations
here so on the way to work we've got
Delta x equals V average T if you
rearrange and solve for T you're going
to get Delta X let's see if I can write
that correctly over V average
which in this case was 60
miles over 40 miles per hour
is going to come out to 60 over 40 the
same thing as 6 over 4 which reduces
down to 3 over 2 or 1.5
hours
all right going home this one you've
probably seen your head if he's
traveling it's an average of 60 miles
per hour then how long does it take him
to go 60 miles well one hour but again
you could set it up just the same way
and just say time is equal to the
distance over the speed the average
speed which is 60 miles over 60
miles per hour which is going to get you
one
hour and then you're simply going to add
these together in one and a half hours
plus one hour is two and a half
hours
cool and that's we're going to plug in
time for the entire time of the round
trip
and so here we've got 120 miles over two
and a half hours and you can definitely
plug this in your calculator you could
also if you wanted to and I like doing
math in my head when I can instead of
two and a half I can write this as five
halves hours so and dividing by five
halves is the same thing as multiplying
by two-fifths so five goes into 120 24
times times two
it's going to get you
48 miles
per hour that is the answer to this
question
cool and as we pointed out before
because he spends more time at 40 miles
per hour than he does at 60 miles per
hour the overall average velocity for
the entire round trip should come out
closer to 40 than it does to 60 and
indeed that's true all right the next
question here is a multi-part question
it says a car accelerates uniformly from
rust with an acceleration of 10 meters
per second squared what is the velocity
after six seconds how far does it travel
in the first second how far does it
travel in the second second how far does
it travel in the third second and the
idea is that if this car is speeding up
then it's going to be going faster and
faster and faster and so how far it
travels every additional second should
be further and further and further so
let's see kind of how we approach this
here so we're told that the uniform
acceleration that acceleration is 10 and
notice it centimeters per second squared
but you might have intuited that I'm
going to write meters per second per
second to make that pretty so it means
it's speeding up 10 meters per second
every second nice round number it's
going to make doing this not so bad at
so first question we want is what is the
velocity
after six seconds so and then we want
how far does it travel in the first the
second and the third second so velocity
after six seconds this you can do in
your head so the acceleration is 10
meters per second per second so we're
told it accelerates from rest which
means the initial velocity is zero it's
a big keyword from rest means your
initial velocity is zero so if you start
at zero and you're speeding up 10 meters
per second every second then one second
after you start you should be moving
with a velocity of 10 meters per second
after two seconds you'll there's an
additional 10 meters per second for that
second second now you're up to 20 meters
per second for the velocity after three
seconds up to 30 meters per second for
the velocity after four seconds up to 40
meters per second for the velocity again
the key is that the acceleration is 10
meters per second every second per
second so every second should go further
the velocity goes up by 10 meters per
second that's how this works and so
you're like well then what is the
velocity after six seconds well if you
started 0 it should be 60 meters per
second and hopefully you can kind of see
that but if you understand what
acceleration means that it's just the
change in the velocity per second and
it's telling you that it's changing by
10 meters per second every second then
after six seconds it'd be 60 meters per
second after 10 seconds would be 100
meters per second after 15 seconds it'd
be 150 meters per second and what you
may not have realized if you're doing it
in your head like that is that you're
technically just using that equation
without actually having to think about
using that equation and if you can think
of acceleration like that that's exactly
where I want you to get and that's why
the problems we're going to do are going
to be nice round numbers because if
instead I'd said instead of saying the
acceleration is 10 meters per second
squared what if I'd said the
acceleration is 1.436 meters per second
squared well now all of a sudden the
math is tough to do in your head you're
going to put out your calculator so and
all of a sudden it's not going to be
this kind of intuitive thing that you're
probably doing in your head but if we
start with nice round numbers then when
you do get harder numbers hopefully
you've got some of those intuitive
Pathways built in and you might still
realize you need to pull out your
calculator but you might recognize what
you need to do in your calculator
without even thinking about it as an
actual equation
so but if we did use that equation
we'd say that the final velocity equals
the initial velocity of zero plus the
acceleration of 10 meters per second
per second times the time
of six seconds and you can see yep that
final velocity after six seconds
is going to be 60 meters per second
notice these seconds cancel and you'll
look for units of meters per second okay
so same thing we just intuitively
calculated now we use an equation to do
it as well
now let's go to the other problems here
and so the question is how far does it
travel in the first second the second
second and so we want Delta X
for zero two one second or maybe we'll
write this a little bit different
so T equals zero
to t equal one second and then we want
the displacement for T equals one second
to T equals two seconds so the second
second and then we want the displacement
for T equals two seconds to T equals
three seconds
cool now there's actually not just one
way of approaching this there's a few
different ways a couple at least a
couple of different ways and we'll go
through a couple of them but we want
displacement so you got to say okay well
which of these equations have
displacement well
the first three all have displacement
that means you're probably using one of
those first three probably not using
that last one so at least not directly
or initially as we'll see let's kind of
take an accounting here and figure out
how we might do this well in this case
first equation Delta x equals V average
T let's write that out
well for that first second how long a
period of time is passing well one
second okay we got the time do we have
the average velocity during that for a
second well we don't but you might
recall again that your average velocity
is equal to V initial plus v final over
two well for that first second at Time
Zero what was the velocity well he
started from rest the velocity was Zero
and after one second we haven't figured
that out yet formally but we did
informally we said the acceleration is
10 meters per second per second so after
one second we said yeah the velocity at
that point would be 10 meters per second
after two seconds it would be 20 meters
per second after three seconds it would
be 30 meters per second and so even
though we haven't formally done it we
can calculate it or just reason it out
in our heads based on knowing what
acceleration is or use this equation to
figure it out and say that V final
equals the initial velocity of zero plus
acceleration of 10 meters per second per
second times one second and you're going
to get a velocity of 10 meters per
second so even though we're not given
what that final velocity is at T equals
one second we can Intuit it or calculate
it very easily and so yeah we don't know
the average velocity off the topper
heads but we can get it pretty quickly
and totally use this so the initial
velocity was Zero the final velocity at
T equals one second is 10 meters per
second so the average is going to come
up to 5 meters per second
and from there
and again from zero to one is one second
and so five meters per second times one
second
gonna get us five meters
so from T equals zero to T equals one
second the displacement is five meters
so what about for the second second from
T equals one second to T equals two
seconds how far now well again this
whatever this object is it is speeding
up and going faster and faster and
faster which means every successive
second we should anticipate that it's
going further and further and further
now we don't know the answer yet but
it's got to be higher than five meters
right
let's figure this out we could do this
the same way and we could rationalize
that on our head so at T equals one
second we already figured out that at T
equals one second
the velocity is 10
meters per second
at T equals two seconds we're going to
be up to
a velocity of 20 meters per second and
if you know the initial and final
velocities then the average is halfway
in between or you can do 10 plus 20 over
2. so and the average would come out to
15 meters per second and we could do the
same thing and say the displacement
equals the average velocity 15 meters
per second and again from T equals one
it's equals two as a total duration of
time of one second so 15 meter second
times one second is 15 meters and we
haven't written anything on the board
yet we just kind of rationalized that
out and did it in our head using again
the exact same equation okay now that's
not the only way to do it but we could
have done it that way and it's 15 meters
it's longer than 5 meters just like we
expected it to be so but let's approach
this a little bit different way let's
see if we can go about this route right
here
and so in this case
Delta x equals V initial t plus one half
a
t squared okay
so in this case v initial well V initial
at T equals one second is 10 meters per
second
I'm sorry yes 10 meters per second
how long is the entire duration of this
journey from T equals one second to
equals took it total of one second
what's our acceleration
10 meters per second per second and
again how long is the entire duration of
this part of the journey again it's a
full one second long
Square it
so by the way big mistake students make
is forgetting the square terms not just
here but another place like equals mc
squared that's another common place
where students forget to actually Square
it they might write it in the equation
then forget to do it in the math so this
is a key common place where that happens
here in physics
so let's look at this out so notice 10
meters per second times one second
that's 10 meters
and then 10 meters per second per second
times one second squared is also going
to come out to 10 meters then times a
half is 5 meters 10 meters plus 5 meters
gets us one other way of getting us our
15 meters
so more than one way to skin a cat so to
speak
all right last one here from two seconds
to three seconds
well from two seconds to three seconds
we can see again that at T equals two
seconds our velocity
equals 20 meters per second by the time
we get to T equals three seconds it'll
have gone up an additional 10 meters per
second based on the acceleration and
should be up to 30.
meters per second
which means the average velocity during
that one second duration somewhere
exactly right halfway in between of 25
meters per second
so and then the whole time is one second
so you say Okay average velocity is 25
meters per second times one second and
it should be 25 meters and again we just
did it in our head or we could use this
equation again and say okay the initial
velocity is 20 meters per second times
again the whole duration from two
seconds to three seconds is one second
so we got 20 meters per second times one
second plus one half times 10 meters per
second per second times one second
squared and you're going to get 20 plus
5 and get 25 meters per second yet again
but maybe there's another way we can do
this
because displacement also shows up in
this equation right here now again this
is my least favorite equation I would
never use this one for this calculation
I just want to demonstrate that it will
lead us to the same answer
okay so in this case we've got
V final squared equals V initial squared
plus two a Delta X so and this one is
kind of the most laborious because again
we've already figured it out that the
time at T equals two seconds the initial
velocity for this period is 20 meter
second the final is 30 meter second
but we had to figure those out and
they're both going to show up here that
we would have to figure those out in
this one as well so alien will get a
square in which makes the math big and
stuff and that's not very fun but that
final velocity was 30 meters per second
and so here we plug that in 30 meters
per second and square it and in fact I'm
going to subtract off this initial
velocity squared as well and it's 20
meters per second
and then we'll square it and that's
going to equal our two times our
acceleration 10 meters per second per
second
foreign
times Delta X here well 30 squared is
920 squared is 400 900 minus 400 is 500.
and then 500 divided by 20 which I might
do differently I might do 500 divided by
the 10. so is 50 and then divided by the
2 is 25 and yet again I get Delta x
equals 25 meters get the same answer
anyway here now it won't always work out
like this like oh you just pick any
other questions it worked out in this
one that we could have used any one of
those three equations
but which one's the easiest and
personally for me
the easiest one for me was again
that guy right there I could pretty
quickly knowing what acceleration
actually means figure out that the
initial velocity at T equals two seconds
so it's 20 meters per second and at T
equals three seconds was 30 meters per
second and at the average is going to be
halfway in between at 25 meters per
second and so during that one second
period 25 meters per second times a
second is 25 meters that was definitely
the easiest way for me but all three of
those equations could have worked
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