Deriving Kinematic Equations - Kinematics - Physics
Summary
TLDRIn this video, you will learn how to derive the four key kinematic equations using a velocity versus time graph. The instructor guides you through the derivation process step-by-step, starting with the relationship between slope and acceleration, then progressing through calculating displacement by analyzing the area under the graph, and finally deriving the equations for velocity and displacement under constant acceleration. The video concludes with a recap of all four equations and a preview of the next lesson, which will cover a useful chart for applying these equations to solve problems.
Takeaways
- 📈 The script explains the derivation of the four kinematic equations using a velocity versus time graph.
- 🔍 The slope of the line on the velocity-time graph represents acceleration, calculated as the change in velocity over time.
- ✏️ The first kinematic equation is derived as \( v_f = v_i + a \cdot t \), where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, and \( a \) is the acceleration.
- 📏 The second equation is derived from the area under the velocity-time graph, representing displacement, and is given by \( \Delta x = v_i \cdot t + \frac{1}{2} a \cdot t^2 \).
- 🔢 The third kinematic equation relates average velocity to displacement and time, and is expressed as \( \Delta x = \frac{v_f + v_i}{2} \cdot t \).
- 🔄 The fourth equation is derived by manipulating the third equation and is written as \( v_f^2 = v_i^2 + 2a \cdot \Delta x \).
- 🔄 The script emphasizes that these equations are applicable for scenarios with constant acceleration.
- 📚 The script mentions that the next video will provide a kinematics chart to help decide which equation to use for different problems.
- 📉 The area under the curve in the velocity-time graph is broken down into a rectangle and a triangle to derive the second kinematic equation.
- 📐 The script uses algebraic manipulation to derive the third and fourth kinematic equations from the first and second equations.
Q & A
What is the first kinematic equation derived from the video?
-The first kinematic equation is \( v_f = v_i + a \cdot t \), where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.
How is acceleration represented on a velocity versus time graph?
-Acceleration is represented as the slope of the line on a velocity versus time graph, calculated as the change in velocity divided by the change in time.
What does the area under the velocity versus time graph represent?
-The area under the velocity versus time graph represents the displacement of an object.
What is the second kinematic equation?
-The second kinematic equation is \( \Delta x = v_i \cdot t + \frac{1}{2} a \cdot t^2 \), which combines the area of a rectangle and a triangle under the velocity-time graph.
How is the average velocity calculated in the context of the third kinematic equation?
-The average velocity is calculated as the displacement divided by the change in time, or equivalently, as the average of the initial and final velocities when acceleration is constant.
What is the third kinematic equation?
-The third kinematic equation is \( \Delta x = \frac{v_f + v_i}{2} \cdot t \), which relates displacement to the average velocity, initial velocity, final velocity, and time.
How is the fourth kinematic equation derived?
-The fourth kinematic equation is derived by manipulating the third equation to solve for \( v_f^2 \), resulting in \( v_f^2 = v_i^2 + 2a \cdot \Delta x \).
What is the significance of the equation \( v_f^2 = v_i^2 + 2a \cdot \Delta x \)?
-This equation is significant as it relates the final velocity squared to the initial velocity squared, acceleration, and displacement, which is useful for problems involving constant acceleration.
Why is it important to consider constant acceleration when using the kinematic equations?
-The kinematic equations are derived under the assumption of constant acceleration. If acceleration is not constant, these equations may not accurately describe the motion.
What is the purpose of a kinematics chart mentioned in the video?
-A kinematics chart helps to decide which kinematic equation to use for a particular problem by providing a visual guide based on the given information in the problem.
What common issue do students face with the kinematic equations, as mentioned in the video?
-Students often struggle to determine which kinematic equation to use for a given problem, which is why a kinematics chart is recommended for assistance.
Outlines
📐 Deriving the First Kinematic Equation
The video begins by explaining how to derive the first of the four kinematic equations using a velocity versus time graph. The straight line on this graph has a slope representing acceleration. Using the slope formula, acceleration is defined as the change in velocity over the change in time. The initial equation is simplified by assuming the initial time is zero, which allows the equation to be manipulated into its final form: V_final = V_initial + at. This is the first kinematic equation, which states that the final velocity equals the initial velocity plus the product of acceleration and time.
🧮 Finding the Second Kinematic Equation Using Area Under the Curve
The second kinematic equation is derived by examining the area under the curve of a velocity versus time graph, which represents displacement. The graph is divided into a rectangle and a triangle. The area of the rectangle (V_initial × time) and the area of the triangle (0.5 × base × height) are summed. By substituting the expression for the height (V_final - V_initial), the formula is refined to V_initial × time + 0.5 × acceleration × time². This results in the second kinematic equation: displacement = V_initial × time + 0.5 × acceleration × time².
📝 Deriving the Third and Fourth Kinematic Equations
The third kinematic equation is introduced by recalling that average velocity is displacement divided by time. For constant acceleration, the average velocity can also be expressed as the sum of initial and final velocities divided by two. By substituting time with (V_final - V_initial) / acceleration, the equation for displacement is expanded and simplified into the third kinematic equation: displacement = (V_initial + V_final) / 2 × time. The video continues to derive the fourth kinematic equation by manipulating the terms, using algebra to connect V_final², V_initial², acceleration, and displacement, resulting in the final form: V_final² = V_initial² + 2 × acceleration × displacement.
📊 Recap of the Four Kinematic Equations
The video recaps the four kinematic equations, ensuring that each is clearly identified and summarized. These equations are used to solve problems involving constant acceleration and include: 1) V_final = V_initial + at, 2) displacement = V_initial × time + 0.5 × acceleration × time², 3) displacement = (V_initial + V_final) / 2 × time, and 4) V_final² = V_initial² + 2 × acceleration × displacement. A brief note is made about common student difficulties in selecting the correct equation for a given problem, with a promise of further guidance in the next video, which will introduce a kinematics chart to assist with equation selection.
Mindmap
Keywords
💡Kinematic Equations
💡Velocity vs. Time Graph
💡Acceleration
💡Displacement
💡Area Under the Curve
💡Initial and Final Velocity
💡Slope Equation
💡Constant Acceleration
💡Average Velocity
💡Kinematics Chart
Highlights
Derivation of the four kinematic equations begins with a velocity versus time graph.
The slope of the line in the velocity versus time graph represents acceleration.
Acceleration is defined as the change in velocity over the change in time.
The first kinematic equation is V_final = V_initial + a*t.
The area under the curve on a velocity-time graph represents displacement.
The second kinematic equation is derived from the area under the curve, resulting in vi*t + (1/2)*a*t^2.
The third kinematic equation involves the average velocity, which is the displacement divided by the change in time.
The average velocity is also expressed as the sum of initial and final velocities divided by 2.
The third kinematic equation is Δx = (V_final^2 - V_initial^2) / 2a.
The fourth kinematic equation is a rearrangement of the third, showing V_final^2 = V_initial^2 + 2aΔx.
The kinematic equations are applicable for problems involving constant acceleration.
A common student challenge is selecting the appropriate kinematic equation for a given problem.
An upcoming video will introduce a kinematics chart to assist in choosing the right equation.
The four kinematic equations are essential for solving motion problems with constant acceleration.
The variables in the kinematic equations represent displacement (Δx), velocity (V), acceleration (a), and time (t).
The video provides a clear explanation of how to derive each kinematic equation step by step.
Understanding the relationship between the area under the velocity-time graph and displacement is crucial.
The video emphasizes the importance of constant acceleration in the application of kinematic equations.
Transcripts
in this video you'll learn how to derive
the four kinematic equations
we'll start with this velocity versus
time graph
on this velocity versus time graph we
have a straight line the slope of this
line represents the acceleration
and then using the slope equation
we can write that the acceleration is
equal to the change in velocity over the
change in time so slope is just the
change in y variable divided by the
change in X variable
from here we can expand that and write V
final minus V initial divided by and
delta T is T final minus t initial but
if the T initial zero we're just going
to go ahead and write t
from here I'm going to move the T on the
bottom right to the left so we get a t
is equal to V final minus V initial
if I add the initial on both sides
I get that V final
is equal to the initial plus a t
oftentimes you'll see it written as it
as it a bit flipped B final equals V
initial plus a t so that is our first
kinematic equation
for the second kinematic equation we're
going to look at the area under the
curve so we're going to look at
this area right here
and the reason we're doing that is
because the area under the curve on a
velocity time graph represents the
displacement
so the area under the curve here we can
see that there is a triangle and a
rectangle so the rectangle the area of
the rectangle is the base times height
so we can take v i times the t v i times
the T so that would be the height times
the base and then here we have a
triangle the area for triangle is one
over two base times height
the height is
the final minus V initial
and then the T times t
from here you'll notice that there's a v
final minus V initial and we saw that
earlier we saw that over here
so we can take this and substitute that
over there
so with that substitution
we get V initial it's time plus one over
two
V final minus V initial is a t
times T and if we multiply that out we
get v i t plus one over two a t squared
and that is our second kinematic
equation
for a third kinematic equation we
remember that the average
velocity is equal to change in
position or the displacement divided by
the change in time
and we also know that
if we were to solve for Delta X
that would be equal to V
the average velocity times the time and
once again the change in time is the
final time minus initial time is the
initial time is zero then you're just
going to have the final time so I'm just
going to call that t
and then next I'm going to
um for the V average I'm going to
substitute that with
this equation V final
plus v initial okay so if if it is any
constant acceleration and which we are
using these kinematic equation for then
the sum of the initial and the final
velocity divided by 2 kind of like if
you want to find the average of two
numbers the average two numbers add them
together divided by 2 and that works for
the initial and final velocity if it's
going to constant acceleration if it's
not then this is not going to work okay
so we're making an important assumption
that we're dealing with constant
acceleration
so and then we have t here
the next step is that you'll notice that
t is equal to if I go let me go over go
back here
over here okay let me come back here
you'll notice that t
if I were switch to T and A I get T is
equal to V final minus V initial divided
by a
and what we're going to do is we're
going to take this
and we're going to substitute this over
here to this T right there okay and what
that gives us is Delta X
is equal to
V final
plus v initial divided by 2
times and then this purple
what's circled on the left on the purple
here is V final minus V initial divided
by a
yeah
all right that looks kind of like a mess
here and then I'm going to keep going so
on the top I've noticed the only
difference is that one is VF plus VI the
other is a minus v f minus VI so we're
doing a little algebra
or you can use foil you know that this
will turn out to be V final squared
minus the initial squared divided by 2A
okay I'm going to go to my next page
so I just copied what I had from the
previous page here so this is where we
left off we have Delta x equals VF
squared minus VI squared divided by 2A
I'm going to move the 2A over so I get
2A Delta x equals to V final squared
minus V initial squared
then I'm going to add VI squared on both
sides VI squared on both sides and that
gives me
VI squared plus 2A Delta x equals V
F squared and often you'll see this
written as VF squared equals v i squared
plus 2A
Delta X that's how you'll usually see
that written and that is your third
kinematic equation
actually this is your fourth kinematic
kinematic equation so let's go back and
just kind of recap the four kinematic
equations I need to point out the third
one I didn't Circle that one but I want
to point that out
all right to recap your four kinematic
equations we have this one
right here which is your your first one
so let me go put a little one right
there this is your second kinematic
equation right here and I didn't Circle
this one but this is actually your third
kinematic equation let me go ahead and
circle that one so this right there
right there that's actually your third
kinematic equation right there and then
we have our fourth kinematic equation
right there okay so let me write all of
these out for you so you can have them
nice and neat all four kinematic
equations
so here are the four kinematic equations
and also on the right hand side I
indicated what the variables stand for
so Delta X is displacement V is velocity
a is acceleration T is time and
typically we'll be using these kinematic
equations where problems are dealing
with constant acceleration
a common issue students have with these
kinematic equations is trying to figure
out which equation to use for a
particular problem so in the next video
I will show you a kinematics chart that
would be really helpful to help you
decide which equation to use to solve a
particular problem
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