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

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, $∆v_x=v_x-v_{x0}$

  • 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, $g=10\mathrm{m}/{\mathrm{s}}^2$

Kinematic equation 1

• This equation is used when position and displacement are not required

$v_x=v_{x0}+a_{x}t$

• Where:

  • $v_x$ = final velocity in the $x$ direction, measured in $\mathrm{m}/\mathrm{s}$

  • $v_{x0}$ = initial velocity in the $x$ direction, measured in $\mathrm{m}/\mathrm{s}$

  • $a_x$ = constant acceleration in the $x$ direction, measured in $\mathrm{m}/{\mathrm{s}}^2$

  • $t$ = time interval, measured in $\mathrm{s}$

Kinematic equation 2

• This equation is used when final velocity is not required

$x=x_0+v_{x0}t+\frac{1}{2}{a}_x{t}^2$

• Where:

  • $x$ = final position in the $x$ direction, measured in $\mathrm{m}$

  • $x_0$ = initial position in the $x$ direction, measured in $\mathrm{m}$

  • $v_{x0}$ = initial velocity in the $x$ direction, measured in $\mathrm{m}/\mathrm{s}$

  • $t$ = time interval, measured in $\mathrm{s}$

  • $a_x$ = constant acceleration in the $x$ direction, measured in $\mathrm{m}/{\mathrm{s}}^2$

Kinematic equation 3

• This equation is used when time is not required

$v_x^2=v_{x0}^2+2a_x\left(x-x_0\right)$

• Where:

  • $v_x$ = final velocity in the $x$ direction, measured in $\mathrm{m}/\mathrm{s}$

  • $v_{x0}$ = initial velocity in the $x$ direction, measured in $\mathrm{m}/\mathrm{s}$

  • $a_x$ = constant acceleration in the $x$ direction, measured in $\mathrm{m}/{\mathrm{s}}^2$

  • $x-x_0$ = displacement, measured in $\mathrm{m}$

Other helpful equations in kinematics

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

$∆x=\frac{1}{2}\left(v_{x0}+v_x\right)t$

• Where:

  • $∆x$ = displacement, measured in $\mathrm{m}$

  • $v_{x0}$ = initial velocity in the $x$ direction, measured in $\mathrm{m}/\mathrm{s}$

  • $v_x$ = final velocity in the $x$ direction, measured in $\mathrm{m}/\mathrm{s}$

  • $t$ = time interval, measured in $\mathrm{s}$

• 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 $\mathrm{m}$

  • $x_0$ = initial position in the $x$ direction, measured in $\mathrm{m}$

  • $v_x$ = final velocity in the $x$ direction, measured in $\mathrm{m}/\mathrm{s}$

  • $t$ = time interval, measured in $\mathrm{s}$

  • $a_x$ = constant acceleration in the $x$ direction, measured in $\mathrm{m}/{\mathrm{s}}^2$

Table of kinematics equations