# Displacement Calculator s = ut + (1/2)at^2

## Calculator Use

This Displacement Calculator finds the distance traveled or displacement (s) of an object using its initial velocity (u), acceleration (a), and time (t) traveled. The equation used is s = ut + ½at^{2}; it is manipulated below to show how to solve for each individual variable. The calculator can be used to solve for s, u, a or t.

## Displacement Equations for these Calculations:

Displacement (s) of an object equals, velocity (u) times time (t), plus ½ times acceleration (a) times time squared (t^{2}).

Where:

s = displacement

u = initial velocity

a = acceleration

t = time

Use standard gravity, a = 9.80665 m/s^{2}, for equations involving the Earth's gravitational force as the acceleration rate of an object.

Different resources use slightly different variables so you might also encounter this same equation with v_{i} or v_{0} representing initial velocity (u) such as in the following form:

Where:

s = displacement

v_{i} = initial velocity

a = acceleration

t = time

## Displacement calculations used in calculator:

Solving for the different variables we can use the following formulas:

- Given u, t and a calculate s

Given initial velocity, time and acceleration calculate the displacement.- s = ut + ½at
^{2}: solve for s

- s = ut + ½at
- Given s, t and a calculate u

Given displacement, time and acceleration calculate the final velocity.- u = s/t - ½at : solve for u

- Given a, u and s calculate t

Given acceleration, initial velocity and displacement calculate the time.- ½at
^{2}+ ut - s = 0 : solve for t using the quadratic formula

- ½at
- Given s, t and u calculate a

Given displacement, time and initial velocity calculate the acceleration.- a = 2s/t
^{2}- 2u/t : solve for a

- a = 2s/t

### Displacement Problem 1:

A car traveling at 25 m/s begins accelerating at 3 m/s^{2} for 4 seconds. How far does the car travel in the 4 seconds it is accelerating?

The three variables needed for distance are given as u (25 m/s), a (3 m/s^{2}), and t (4 sec).

s = ut + ½at^{2}

s = 25 m/s * 4 s + ½ * 3 m/s^{2} * (4 s)^{2}

s = 100 m + 0.5 * 3 m/s^{2} * 16 s^{2}

s = 100 m + 0.5 * 48 m

s = 100 m + 24 m

**s = 124 meters**

You can check this answer with the Math Equation Solver: 25 * 4 + 0.5 * 3 * 4^2

### Displacement Problem 2:

It takes a plane, with an initial speed of 20 m/s, 8 seconds to reach the end of the runway. If the plane accelerates at 10 m/s^{2}, how long is the runway?

s = ut + ½at^{2}

s = 20 m/s * 8 s + ½ * 10 m/s^{2} * (8 s)^{2}

s = 160 m + 0.5 * 10 m/s^{2} * 64 s^{2}

s = 160 m + 0.5 * 640 m

s = 160 m + 320 m

**s = 480 meters**

You can check this answer with the Math Equation Solver: 20 * 8 + 0.5 * 10 * 8^2