Kinematic Equations

Kinematic equations is a list of formulas used to describe the motion of objects, in particular, when acceleration is a constant.

In the following equations, $x$ = displacement (distance), $v$ = velocity, $a$ = acceleration, and $t$ = time. A variable with a subscript $i$, like $v_{i}$ or $d_{i}$ represents the initial value of that variable.

$v = v_{i} + at$
$x = \dfrac{v + v_i}{2}t$
$x = v_i t + \frac{1}{2}at^2$
$v^2 = v_i^2 + 2ax$

Which equation to use?

To determine which equation you should use, first determine which information you have and which piece of information you are solving for. Then, find the formula that has only those parts in it.

Example
A car accelerates at a constant rate from rest to 72 ft/s over 0.24 mi. How long did it take the car to reach 72 ft/s?
s
Solution

In this problem, we know the initial velocity of the car (0 ft/s), the final velocity of the car (72 ft/s), and the distance it took the car to reach the final velocity (0.24 mi). We are solving for the amount of time the car traveled to reach the final velocity. The equation that has all of these parts is $x = \dfrac{v + v_i}{2}t$.

You can substitute the values into this equation and solve for $t$ or you can solve for $t$ algebraically first, and then substitute values in.

If we solve for $t$ first, we get $t = \dfrac{2x}{v + v_i}$. Substituting, we get $t = \dfrac{2(0.24\ \text{mi}) \times \frac{5280\ \text{ft}}{1\ \text{mi}}}{0\ \text{ft/s} + 72\ \text{ft/s}} = 35.2\ \text{s}$.