# Kinematic equations in one dimension

### Kinematic equations in one dimension

#### Lessons

In this lesson, we will learn:

• The four kinematic equations
• How to choose which kinematic equation to use
• Problem solving with the kinematic equations

Notes:

The four kinematic equations describe the relationship of the initial velocity ($v_{i}$), final velocity ($v_{f}$), acceleration ($a$), displacement ($d$), and time ($t$) for an object moving in one dimension. Each of the equations is made up of four of the five of these variables. If we know three of these variables, we can use the kinematic equations to solve for the two remaining unknown variables.

Kinematic Equations
1. $v_{f}=v_{i}+at$(No $d$)
2. $v_{f}^{2}=v_{i}^{2}+2ad$(No $t$)
3. $d=v_{i}t+\frac{1}{2}at^{2}$(No $v_{f}$)
4. $d=(\frac{v_{i}+v_{f}}{2})t$(No $a$)

$v_{i}$: initial velocity, in meters per second (m/s)

$v_{f}$: final velocity, in meters per second (m/s)

$a$: acceleration, in meters per second squared $(m/s^{2})$

$t:$ time, in seconds (s)

$d:$ displacement, in meters (m)

• Introduction
Introduction to kinematic equations
1. Learn how to translate a question into kinematic terms: $v_{i}, v_{f}, a, t, d$
2. Learn how to select an applicable kinematic equation to solve for the unknown.

• 1.
Applying kinematics equations to horizontal motion
a)
A sprinter accelerates from rest to 12.4 m/s in 9.58 s. What is the average acceleration of this sprinter?

b)
A commercial airplane must reach a speed of 75.0 m/s for takeoff. How long of a runway is needed if the acceleration is 3.20 m/s$^{2}$?

c)
A car traveling 15.0 m/s goes uphill with a uniform acceleration of -1.80 m/s$^{2}$. How far has it traveled after 5.00 s?

d)
How long does it take a car to decelerate from 85.0 km/h to 50.0 km/h in 100 m?

• 2.
Applying kinematics equations to vertical motion
a)
How long does it take a ball to hit the ground if it is dropped from a height of 1.50 m:
1. on Earth, where acceleration due to gravity is 9.80 m/s$^{2}$?
2. on the moon, where acceleration due to gravity is 1.62 m/s$^{2}$?

b)
A rock is thrown straight up with a speed of 20.0 m/s. How high does it go?

c)
A rock is thrown straight up with a speed of 20.0 m/s. How long is it in the air?

d)
A rock is thrown straight up with a speed of 20.0 m/s. When does the rock have a height of 15.0 m?

e)
A rock is thrown straight up with a speed of 20.0 m/s, and is caught when it comes back down.
1. What is the velocity of the rock when it is caught?
2. What is the velocity of the rock when it is at a height of 15.0 m? Include both $v$ as it goes up and $v$ as it comes back down.

f)
A ball is tossed straight up into the air with a speed of 12.0 m/s from the edge of a 324 m tall building.
1. What is the impact speed of the ball as it hits the ground?
2. What total distance did the ball travel?

• 3.
Solving "two part" motion in one dimension
a)
A car travels at 23.5 m/s when the driver sees a deer on the road 55.0 m ahead of his car. It takes the driver 0.300 s of reaction time to hit the brakes, and the car starts decelerating at 6.40 m/s$^{2}$. Will the car stop in time to avoid hitting the deer?

b)
A falling stone takes 0.330 s to travel past a window 2.20 m tall. From what height above the top of the window did the stone fall?