Chapter 2 Part 3 Four Kinematic Equations

Prof. Clements Notes/Hints for Physics Courses

Chapter 2 Open Stax College Physics or Most Introductory Physics Courses

Motion Equations, Kinematics

I highly recommend that you go to the web site https://openstax.org/details/books/college-physics to access the free OpenStax College Physics textbook and resources.

YouTube videos of my (usually short) lectures and example problems are indexed at http://www.physics.gpclements.com/ . There are also a few videos that give review of basic math tools that are used in an introductory physics course.

Terms to know: constant acceleration, kinematic equations

Motion Equations The early part of this chapter in most textbooks describes position, velocity, and acceleration. Now these quantities will be connected in equations. There are four equations of motion that we will use. You should be familiar with the book’s notation and my notation. There may be some slight differences in our notation. The time value at the start of the interval will be taken to be zero. The final time is labeled t. Any symbol with a subscript 0 is a quantity at the start of the time interval (often time = 0). The quantities without a subscript are the final values.

If the value of the acceleration changes during a problem then you must work the problem in separate steps, with the acceleration constant during each step.

If the acceleration is constant the acceleration may be some positive number or zero or some negative numbers. “Constant acceleration” does not automatically mean that the acceleration is zero. Also, some students have a misconception that if the acceleration is constant then the value of the velocity is zero. The previous statement is not true.

If the acceleration is a constant +5 m/s² , then every second the velocity will increases by 5 m/s. i.e. Suppose the velocity is +3 m/s at time = 0. After one second the velocity would be +8 m/s. After two seconds the velocity would be +13 m/s. As a check, calculate the average acceleration a = (V – V_o ) / t using the t=0 and t=2 seconds data.

There are two basic equations of motion. Using these two equations we can apply some algebra and develop two more useful equations. There are four useful kinematic equations of motion. The term kinematic tells us that we are not worrying about the cause of motion with these equations. The kinematic equations just describe motion.

The following development of the four equations may be in a slightly different order than your textbook but we will end up with the same equations of motion.

1^st Kinematic Equation: By definition a = V / t or a = (V - V_o ) / t Multiply both sides by t and add V_o to both sides.

Equation 1 V = V_o + at

Suppose the acceleration is 4 m/s², the initial velocity is +3 m/s, and the initial position is 7 meters. What is the velocity when t = 5 seconds? Answer: V = V_o + at

V = 3 m/s + 4 m/s² * 5 sec V = 23 m/s

2^nd Kinematic Equation: The next equation lets us calculate the final position if we know the initial position, the average velocity and the time. For the case of constant acceleration It turns out that the average velocity can be found by adding the initial and final velocities and then dividing by 2. The displacement is equal to the average velocity multiplied by time: X - X_o = [ ½ (V + V_o) ] * t or

X = X_o + [ ½ (V + V_o) ] * t I will refer to this as Equation 2.

Your textbook may use this form S = [ ½ (V + V_o) ] * t Here S = X - X_o

This equation can also be derived by analyzing the graph of V vs t as shown in this image.

X = X_o + ½ (V + V_o) t    This is only valid when the acceleration is constant because the average velocity will not equal ½ (V + V_o) if the acceleration is not constant.

Let us continue to use the data given in the example calculation done for Kinematic Equation 1: X_o = 7 meters V_o = 3 m/s V = 23 m/s a = 4 m/s²

What is the position of the object at t = 5 seconds? Answer: X = X_o + ½ (V + V_o) t

X = 7 meters + 0.5 ( 3m/s + 23 m/s) 5 seconds

X = 7 meters + 0.5 ( 26 m/s) 5 seconds

X = 7 meters + 13 m/s *5 seconds Note * indicates multiplication

X = 7 meters + 65 meters X = 72 meters

3^rd Kinematic Equation: We can combine the first two equations by substituting for V in equation 2 with the expression for V from equation 1. This will generate an equation that does not contain the final velocity. Start with equation 2 X = X_o + ½ (V + V_o) * t

Replace V with the expression from equation 1 X = X_o + ½ ( [V_o + at] + V_o) * t

You should use the principles of algebra to simplify this. Look at the image at the end of this document for help. And, ask your instructor if you don’t obtain:

Equation 3 X = X_o + V_o * t + 0.5 * a * t² or S = V_o * t + 0.5 * a * t²

e.g. Suppose the acceleration is 4 m/s², the initial velocity is +3 m/s, and the initial position is 7 meters. Find the position at time = 5 seconds. Treat the final velocity as unknown.

X = 7 m + 3 m/s * 5 sec + 0.5 * 4 m/s² * ( 5 sec )²

X = 7 m + 15 m + 50 m

X = 72 m Does this result sound familiar?

It is worth examining equation 3 for the case of acceleration = 0. With a = 0 the equation becomes X = X₀ + V₀ * t or (X - X₀ ) = V₀ * t You may recognize this as being similar to your math problems involving Distance = rate * time. D = rt

D = rt is only valid if the rate is constant. (i.e. acceleration = 0, and V = V₀ )

e.g. Initial position = 0 meters, a = 0, initial velocity +4 m/s, time = 3 seconds

D = 4 m/s * 3 seconds = 12 meters

4^th Kinematic Equation: We can combine the first two equations by solving for time in equation 1 and substituting this expression for t into equation 2. This is done in the image at the end of this document.

Equation 4 V² = V_o² + 2 * a * (X – X_o) or V² = V_o² + 2 * a * S

e.g. Suppose the acceleration is 4 m/s², the initial velocity is +3 m/s, the initial position is 7 meters, and the final position is 72 meters. Find the final velocity. Treat time as unknown.

V² = (3 m/s)² + 2 * 4 m/s² * ( 72 meters – 7 meters)

V² = 9 m²/s² + 8 m/s² * 65 m

V² = 9 m²/s² + 520 m²/s² or V² = 529 m²/s²

Take the square root of both sides and you find V = 23 m/s . You should compare this result to the final velocity calculated above for the example for Kinematic Equation 1.

You now know the 4 Kinematic Equations. These equations only give correct results when the acceleration is constant but that covers many situations that you will study in your physics class. The law of Conservation of Energy (future chapter) will help us solve problems when the acceleration is not constant.

You should work through some of the examples in your textbook until you are comfortable using these equations. If there is an example that doesn’t make sense ask your instructor to go over the calculations steps.

Problem-Solving for One-Dimensional Kinematics

Read through the suggested steps for problem solving in your textbook. I strongly recommend that you start by reading the entire problem. Then read it again and make a sketch of the situation. The sketch does not need to be artistic. Sketch the objects (stick figures, boxes, circles, etc.) and select some coordinate system. Label 0 on your coordinate system. Use arrows to indicate velocities and accelerations of objects. Write down the numeric values next to the symbols (X_o, X, V_o , V, a, t ) used in the 4 kinematic equations. For simpler problems you may be able to select one of the four kinematic equations such that there is only one unknown.

For some problems you may have to use more than one equation before you arrive at the final answer.

It is also very important to check that your answer is reasonable. For some situations you won’t know what is reasonable (e.g. speed of an electron). Do your work by yourself and then check with a class mate to see if they have the same answer. Ask your instructor for guidance on rounding or not rounding intermediate values in the calculations. I would suggest that you store intermediate results in the memory of your calculator and don’t round off numbers before you reach the final result. Ask your instructor if you should follow the rules of significant figures for homework and exam problems. In the absence of directions on this I would suggest you write down three digits in your answer. e.g. 5.26 x 10³ m/s , 4.68 m/s² , etc.

The following image shows the derivation of the 4 Kinematic Equations:

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.