Deriving Kinematic Equations - Kinematics - Physics

Summary

In 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

00:00

📐 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.

05:04

🧮 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.

Keywords

💡Kinematic Equations

Kinematic equations are fundamental formulas in physics that describe the motion of an object under constant acceleration. In the video, the derivation of these equations is the central theme, as they are used to relate displacement, velocity, acceleration, and time. The equations are derived from a velocity versus time graph, and they are essential for solving problems in classical mechanics.

💡Velocity vs. Time Graph

A velocity versus time graph is a graphical representation of an object's velocity changes over time. In the script, this graph is used to introduce the concept of acceleration, which is the slope of the line on the graph. The graph is crucial for deriving the first kinematic equation, where the slope represents the rate of change of velocity with respect to time.

💡Acceleration

Acceleration is defined as the rate of change of velocity with respect to time. It is a vector quantity that indicates how quickly the velocity of an object changes. In the video, acceleration is derived from the slope of the velocity-time graph, and it is a key component in the kinematic equations, as it helps determine how an object's velocity changes over time.

💡Displacement

Displacement refers to the change in position of an object. In the context of the video, displacement is calculated by the area under the velocity-time graph. The script explains that the area under the curve, which includes a rectangle and a triangle, represents the total displacement of the object, which is a crucial component in the second kinematic equation.

💡Area Under the Curve

The area under a velocity-time graph represents the total displacement of an object. In the script, the area is calculated by summing the areas of a rectangle and a triangle, which correspond to different phases of motion. This concept is used to derive the second kinematic equation, which relates displacement to initial velocity, final velocity, and acceleration.

💡Initial and Final Velocity

Initial and final velocities are the velocities of an object at the beginning and end of a time interval, respectively. These velocities are essential in the kinematic equations to calculate changes in velocity and to determine the object's motion. The script uses these velocities to derive equations that relate to the object's acceleration and displacement.

💡Slope Equation

The slope equation is a mathematical formula used to calculate the slope of a line, which in physics represents the rate of change of one variable with respect to another. In the video, the slope equation is used to relate acceleration to the change in velocity over time, which is the basis for the first kinematic equation.

💡Constant Acceleration

Constant acceleration implies that the rate of change of velocity is unchanging over time. This assumption is crucial for the validity of the kinematic equations. The video script emphasizes that these equations are applicable when the acceleration is constant, which simplifies the analysis of motion.

💡Average Velocity

Average velocity is calculated as the total displacement divided by the total time taken. In the script, average velocity is used to derive one of the kinematic equations, which relates the average velocity to the total displacement and time. This concept is important for understanding motion when acceleration is not constant.

💡Kinematics Chart

A kinematics chart is a tool used to help select the appropriate kinematic equation for a given problem. The video script mentions that such a chart will be introduced in a subsequent video to assist students in applying the kinematic equations effectively to solve problems involving motion under constant acceleration.

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