Kinematic Equations (College Board AP® Physics 1: Algebra-Based)

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Leander Oates

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Physics

Kinematic equations

In physics, motion can be represented in a number of different ways:
- Motion diagrams
- Figures and data
- Graphs
- Equations
- Narrative descriptions
For objects in a state of constant or uniform acceleration, there are three kinematic equations that can be used to describe their instantaneous linear motion in one dimension
- Each of the three kinematics equations is listed on the equation sheet and will be provided in the exams
For all of these equations, the following conditions apply:
- acceleration is uniform
  - therefore, average and instantaneous acceleration are equal
- motion is along a straight line
  - motion is presented in the $x$ direction but can also be applied to the $y$ direction
For all these equations:
- Time interval, $∆ t = t$
  - The assumption is that the timer is started from zero, $t_{0} = 0$
- Change in velocity, $∆ v_{x} = v_{x} - v_{x 0}$
- Displacement, $∆ x = x - x_{0}$
In exam questions, some of the information required for the calculation may be assumed
Phrases and situations to look out for:
- The object begins at rest
  - The object is initially stationary; therefore, the initial velocity is zero
  - This leads to any expression containing initial velocity canceling to zero
- Falling objects or objects in free fall
  - The acceleration is equal to the acceleration of free fall at Earth's surface, $g = 10 m / s^{2}$

Kinematic equation 1

This equation is used when position and displacement are not required

$v_{x} = v_{x 0} + a_{x} t$

Where:
- $v_{x}$ = final velocity in the $x$ direction, measured in $m / s$
- $v_{x 0}$ = initial velocity in the $x$ direction, measured in $m / s$
- $a_{x}$ = constant acceleration in the $x$ direction, measured in $m / s^{2}$
- $t$ = time interval, measured in $s$

Kinematic equation 2

This equation is used when final velocity is not required

$x = x_{0} + v_{x 0} t + \frac{1}{2} a_{x} t^{2}$

Where:
- $x$ = final position in the $x$ direction, measured in $m$
- $x_{0}$ = initial position in the $x$ direction, measured in $m$
- $v_{x 0}$ = initial velocity in the $x$ direction, measured in $m / s$
- $t$ = time interval, measured in $s$
- $a_{x}$ = constant acceleration in the $x$ direction, measured in $m / s^{2}$

Kinematic equation 3

This equation is used when time is not required

$v_{x}^{2} = v_{x 0}^{2} + 2 a_{x} (x - x_{0})$

Where:
- $v_{x}$ = final velocity in the $x$ direction, measured in $m / s$
- $v_{x 0}$ = initial velocity in the $x$ direction, measured in $m / s$
- $a_{x}$ = constant acceleration in the $x$ direction, measured in $m / s^{2}$
- $x - x_{0}$ = displacement, measured in $m$

Other helpful equations in kinematics

Displacement can be calculated using velocity and time when acceleration is not required

$∆ x = \frac{1}{2} (v_{x 0} + v_{x}) t$

Where:
- $∆ x$ = displacement, measured in $m$
- $v_{x 0}$ = initial velocity in the $x$ direction, measured in $m / s$
- $v_{x}$ = final velocity in the $x$ direction, measured in $m / s$
- $t$ = time interval, measured in $s$
Final position can be calculated when initial velocity is not required using the following equation:

$x = x_{0} + v_{x} t - \frac{1}{2} a_{x} t^{2}$

Where:
- $x$ = final position in the $x$ direction, measured in $m$
- $x_{0}$ = initial position in the $x$ direction, measured in $m$
- $v_{x}$ = final velocity in the $x$ direction, measured in $m / s$
- $t$ = time interval, measured in $s$
- $a_{x}$ = constant acceleration in the $x$ direction, measured in $m / s^{2}$

Table of kinematics equations

Equation	Quantity not required
$v_{x} = v_{x 0} + a_{x} t$	$(x - x_{0})$
$x = x_{0} + v_{x 0} t + \frac{1}{2} a_{x} t^{2}$	$v_{x}$
$v_{x}^{2} = v_{x 0}^{2} + 2 a_{x} (x - x_{0})$	$t$
$∆ x = \frac{1}{2} (v_{x 0} + v_{x}) t$	$a_{x}$
$x = x_{0} + v_{x} t - \frac{1}{2} a_{x} t^{2}$	$v_{x 0}$

Examiner Tip

It can be quicker to choose a kinematic equation based on the quantity that is not required for the calculation. The table above is a handy way to locate the relevant equation based on the quantity not required.

Worked Example

A rock is dropped from a bridge and strikes the water with an impact velocity of $18 m / s$ .

Calculate the height of the bridge.

Answer:

Step 1: Check for any implied quantities

The rock is dropped; this implies that the initial velocity is zero
The rock is in free fall; this implies that the acceleration is equal to the acceleration due to gravity at Earth's surface

Step 2: List the known quantities

Taking the positive direction to be downward
Initial velocity, $v_{y 0} = 0 m / s$
Final velocity, $v_{y} = 18 m / s$
Uniform acceleration, $a_{y} = 10 m / s^{2}$

Step 3: Choose the relevant kinematic equation

The question asks for the height of the bridge, which is equal to the displacement of the rock
The quantity not required in this calculation is time

$v_{y}^{2} = v_{y 0}^{2} + 2 a_{y} (y - y_{0})$

Step 4: Check if any of the quantities cancel to zero

Initial velocity is zero, therefore:

${v^{2}}_{y} = 0 + 2 a_{y} (y - y_{0})$

${v^{2}}_{y} = 2 a_{y} (y - y_{0})$

Step 5: Rearrange the equation

The question asks for displacement, so make $(y - y_{0})$ the subject

$(y - y_{0}) = \frac{v_{y}^{2}}{2 a_{y}}$

Step 6: Substitute in the known values to calculate

$∆ x = \frac{18^{2}}{2 \cdot 10}$

$∆ x = 16.2 m$

Examiner Tip

There is only one numerical value given in the question above; two of the values are implied. At first glance, it may seem like you don't have enough information to solve the problem. This should be seen as a clue that there is extra information hiding in the wording of the question, or that you may have already calculated a value in a previous question part.

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Kinematic Equations (College Board AP® Physics 1: Algebra-Based)

Study Guide

Kinematic equations

Kinematic equation 1

Kinematic equation 2

Kinematic equation 3

Other helpful equations in kinematics

Table of kinematics equations

Examiner Tip

Worked Example

Examiner Tip

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Author: Leander Oates