How to Remember/Derive the Kinematics Equations
TLDRThis script offers an insightful breakdown of the four fundamental kinematic equations used in physics, focusing on how to memorize and derive them effectively. It explains the relationship between velocity, acceleration, displacement, and time, using clear examples and graphical representations. The video simplifies complex concepts by demonstrating how to rearrange and combine the equations to solve for various variables, ultimately emphasizing the importance of understanding these principles for a solid grasp of kinematics.
Takeaways
- 📚 The four fundamental kinematic equations are essential for understanding physics.
- 🧠 Memorizing these equations can be facilitated by understanding and applying pre-existing knowledge.
- 🔄 The first kinematic equation relates final velocity (v) to initial velocity (u), acceleration (a), and time (t): v = u + at.
- 📈 Acceleration is the slope of the velocity vs. time graph, which can be expressed as a = (v - u) / t.
- 🔄 The second equation describes displacement (Δx) as the average velocity (v_avg) times time (t): Δx = v_avg * t, where v_avg = (u + v) / 2.
- 🏠 Displacement can also be visualized as the area under the velocity vs. time graph, calculated as (base * height) / 2.
- 🔄 The third equation connects displacement (Δx) with initial velocity (u), acceleration (a), and time (t): Δx = u * t + 1/2 * a * t^2.
- 🔄 By substituting known equations, we can derive that the final velocity squared (v^2) equals the initial velocity squared (u^2) plus twice the product of acceleration and displacement (2aΔx).
- 🔄 The fourth equation is derived by eliminating the time variable and is expressed as: (v^2 - u^2) = 2aΔx.
- 📊 The kinematic equations can be visualized and proven using graphs, such as the velocity vs. time graph.
- 💡 Understanding and applying these kinematic equations is crucial for solving problems in physics involving motion and displacement.
Q & A
What are the four kinematic equations discussed in the transcript?
-The four kinematic equations discussed are: 1) Final velocity = initial velocity + acceleration × time, 2) Displacement (ΔX) = (V₀ + Vₕ) / 2 × time (T), 3) ΔX = V₀ × T + 0.5 × a × T², and 4) Vₕ² = V₀² + 2a(ΔX).
How is acceleration defined in the context of the first kinematic equation?
-Acceleration is defined as the slope of a velocity versus time graph, or equivalently, as the change in velocity over the change in time (acceleration = ΔV / ΔT).
What does the second kinematic equation indicate about average velocity?
-The second kinematic equation indicates that displacement is equal to the average velocity times time. The average velocity is calculated as (initial velocity + final velocity) / 2.
How can you derive the third kinematic equation from the first two?
-By eliminating the final velocity (Vₕ) from the first two equations and rearranging the terms, you can derive the third equation: ΔX = V₀ × T + 0.5 × a × T².
What is the significance of the fourth kinematic equation, Vₕ² = V₀² + 2a(ΔX)?
-The fourth kinematic equation relates the final velocity squared to the initial velocity squared, acceleration, and displacement. It is useful for solving problems where the relationship between these quantities is needed.
How does the transcript explain the concept of change in a variable?
-The transcript explains the concept of change as the difference between the final and initial values of a variable, such as change in velocity (ΔV) being final velocity minus initial velocity.
What is the method used in the transcript to help memorize the kinematic equations?
-The method used is to understand and derive the equations using known concepts and relationships, such as the slope of a graph and the average of two numbers, to build a deeper understanding and facilitate memorization.
How can the area under a velocity vs. time graph be used to find displacement?
-The area under the velocity vs. time graph can be used to find displacement by calculating the area of the shape formed by the graph, the time axis, and the initial velocity line. For example, if the shape is a triangle, the displacement is equal to (base × height) / 2.
What is the role of the distributive property in simplifying the third kinematic equation?
-The distributive property is used to simplify the third kinematic equation by breaking it down into simpler terms, which allows for easier manipulation and substitution to derive the final form of the equation (ΔX = V₀T + 0.5aT²).
How does the transcript demonstrate the process of solving for time in the second kinematic equation?
-The transcript demonstrates solving for time by isolating it on one side of the equation. By dividing both sides of the equation ΔX = (V₀ + Vₕ) / 2 × T by (V₀ + Vₕ) / 2, time (T) is found to be equal to 2ΔX / (V₀ + Vₕ).
What is the final step in deriving the fourth kinematic equation?
-The final step in deriving the fourth kinematic equation is to add initial velocity to both sides of the derived equation (Vₕ² - V₀² = 2aΔX) to get the final form: Vₕ² = V₀² + 2aΔX.
Outlines
📚 Introduction to Kinematics Equations
This paragraph introduces the four fundamental kinematics equations essential for understanding physics. It emphasizes the importance of memorizing these equations and presents a method to derive them using known variables. The first equation discussed is the final velocity equation, which relates initial velocity, acceleration, and time. The explanation includes the concept of acceleration as the slope of a velocity-time graph and the derivation of the equation through rearranging known relationships.
📈 Derivation and Application of Displacement Equations
The second paragraph delves into the derivation and application of displacement equations. It explains the concept of average velocity and how it relates to displacement, using a velocity-time graph for illustration. The paragraph provides a step-by-step breakdown of how displacement can be calculated as the area under the velocity-time curve. It also demonstrates how to derive the equation for displacement in terms of initial velocity, acceleration, and time by combining and simplifying the previously discussed equations.
Mindmap
Keywords
💡Kinematics
💡Velocity
💡Acceleration
💡Time
💡Displacement
💡Average Velocity
💡Change in Velocity
💡Slope
💡Graph
💡Area Under the Line
💡Distributive Property
Highlights
Four kinematics equations are discussed, providing a foundational understanding for physics.
The method to memorize kinematics equations efficiently is introduced, building on pre-existing knowledge.
The first kinematics equation relates final velocity, initial velocity, acceleration, and time.
Acceleration is defined as the slope of a velocity-time graph, offering a geometric interpretation.
Change in velocity is quantified as the difference between final and initial velocities.
The second kinematics equation connects displacement, average velocity, and time, with a practical example provided.
Displacement is calculated as the area under the velocity-time graph, demonstrated with a specific example.
The third kinematics equation is derived by eliminating the final velocity, showcasing algebraic manipulation.
The relationship between displacement, initial velocity, acceleration, and time is further explored and simplified.
The fourth kinematics equation is derived by eliminating the time variable, highlighting the power of substitution and algebraic simplification.
A comprehensive understanding of kinematics equations is achieved through the derivation and application of these fundamental relationships.
The video content serves as a valuable resource for anyone needing to understand or review kinematics equations.
The use of graphical representations, such as the velocity-time graph, aids in the visualization and comprehension of the concepts.
The practical application of these equations is emphasized through the use of real-world examples and scenarios.
The video provides a step-by-step approach to deriving and understanding the kinematics equations, making it accessible to a wide range of learners.
The final equation derived, relating final velocity squared to initial velocity squared, acceleration, and displacement, encapsulates the essence of the kinematics discussion.
The transcript serves as a detailed guide for anyone looking to deepen their understanding of the fundamental principles of physics.
Transcripts
so these are the four kinematics
equations that you need to know for
physics and yeah that's a lot of
variables so I'll be showing you in this
video how you can memorize them
efficiently and use what you already
know to derive them so our first one is
velocity or final velocity equals
initial velocity plus acceleration times
time so we already know that
acceleration is the slope of a velocity
versus time graph in other words
acceleration equals change in velocity
over change in time and we also know
that change in velocity is equal to
final minus initial because that's what
change in really stands for for anything
so a change in position would be final
position minus initial position so now
we know this now that we know this we
have acceleration equals final velocity
minus initial velocity over time
well that's starting to look a lot like
what we have here because when we
rearrange this and multiply a time by
both sides to get acceleration times
time equals final minus initial velocity
and if we add over the initial velocity
over here we get a T plus V sub 0 equals
final velocity and that's essentially
what we have here final velocity equals
initial plus acceleration times time ok
so our second one is displacement or
Delta X which is change in position
equals V sub 0 plus V sub F over 2 times
T ok well we know that the average of
two things are is that one thing plus
another thing over 2 so this is
basically equal to average velocity so V
average equals initial velocity plus
final velocity over 2 because initial
velocity is our first thing final
velocities are a second thing and we're
dividing it by 2 which is
the number of numbers that we have so
this is basically saying that
displacement is equal to average
velocity times time and you can prove
that using a velocity versus time graph
so we have velocity which is measured in
meters per second on the y-axis and time
which is measured in seconds on the
x-axis so let's say we have two points 0
0 and another point 3 1 point 5 so after
3 seconds the velocity is 1.5 meters per
second and displacement is equal to the
area under this line so it would be the
this triangle area so we'll say that
displacement equals the base times
height over 2 so this would be time
equals 3 seconds so the base is 3 and
our height is 1.5 and we're multiplying
that by 1/2 so 3 times 1.5 is 4 point 5
divided by 2 is 2 point 2 5 so our
displacement is 2.25 meters well this
also works when we take the average
velocity and multiply it by time so what
is average velocity well we have a
velocity of 0 for our initial velocity
and a velocity of 1.5 for our final
velocity so 0 plus 1.5 over 2 is going
to equal our average velocity otherwise
known as 0.75 meters per second so when
you multiply 0.75 which is our V average
times T which is 3 seconds we get 0.75
times 3 which is also equal to 2 point 2
5 meters and the third one is
displacement so Delta x equals 1/2 a T
squared plus initial velocity times time
so V sub 0 times T and we can get this
by using the two equations that we have
already so I see here that we only have
an initial velocity so our final
velocity is what's eliminated in this
equation so let's try to eliminate V sub
F and we already know that V sub F
equals initial velocity plus
acceleration times time so we can
substitute this with what we have here
so let me rewrite this by saying Delta X
equals initial velocity plus final
velocity which is this so initial
velocity plus acceleration times time
over 2 multiplied by time okay and I see
that these two are the same thing so
Delta X equals to initial velocity plus
acceleration times time over 2
multiplied by time and we can further
simplify this by breaking it up into
fractions so Delta X equals this so to
floss initial velocity over 2 plus
acceleration times time over 2 and all
of this is multiplied by time okay well
we can further simplify this by crossing
out those twos so Delta X equals initial
velocity and don't forget to multiply at
that time so distributive property so
initial velocity times time plus 1/2
which is coming from this divided by 2
and then a T squared because we're
multiplying by a T times another T so
that's going to give us a T squared and
look at that that looks exactly like
this equation we have here so our final
equation is final velocity squared
equals initial velocity squared plus two
times acceleration times displacement so
let me just rewrite all of that to a
delta X and we're going to use these two
equations again to get this fourth one
and in this fourth one I don't see a
time variable here so we'll have to
eliminate time so let's solve for time
using the second equation then and the
second equation is Delta x equals V sub
0 plus P sub F over 2 times time ok now
let's solve for time by dividing both
sides by this whole thing
so we get time equals Delta X which is
on top divided by this whole thing which
is V sub 0 plus V sub F over 2 which
simplifies into 2 Delta X over V sub 0
plus V sub F ok so now that we know what
time is we can substitute it into this
first equation which is final velocity
equals initial velocity plus
acceleration times time ok so now that
we know what time is we can plug it into
this equation here so V sub F equals
initial velocity plus acceleration times
2 Delta X over V sub 0 plus V sub that's
ok
that's what now let's make it look a
little more like this equation by first
subtracting initial velocity from both
sides so we have final minus initial
equals and then we can multiply this out
so 2a Delta X over industrial velocity
plus final velocity ok we're almost
there
so now we all we need to do is multiply
both sides by the denominator so we can
cancel it out so we have final minus
initial
which is this thing multiplied by this
thing which is initial velocity plus
final velocity equals to a delta X okay
now is almost the second-to-last part
actually it is the second-to-last part
so we have V sub F times V sub 0 plus V
sub F times V sub F which is V sub F
squared minus initial velocity squared
from multiplying these two and then
minus V sub 0 P sub F okay and this
thing V sub 0 times V sub F is the same
thing as V sub F times V sub 0 so they
cancel each other out so now we have
final velocity squared minus initial
velocity squared equals to a delta X and
if we add initial velocity to both sides
we get final velocity squared equals
initial velocity squared plus 2 a delta
X wow that was a lot but we finally got
all of our kinematics equations
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