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 + ½at2; 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 (t2).

\( s = ut + \dfrac{1}{2}at^2 \)

Where:
s = displacement
u = initial velocity
a = acceleration
t = time

Use standard gravity, a = 9.80665 m/s2, 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 vi or v0 representing initial velocity (u) such as in the following form:

\( s = v_it + \dfrac{1}{2}at^2 \)

Where:
s = displacement
vi = 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 + ½at2: solve for s
• 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.
  • ½at2 + ut - s = 0 : solve for t using the quadratic formula
• Given s, t and u calculate a
  Given displacement, time and initial velocity calculate the acceleration.
  • a = 2s/t2 - 2u/t : solve for a

Displacement Problem 1:

A car traveling at 25 m/s begins accelerating at 3 m/s2 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/s2), and t (4 sec).

s = ut + ½at2
s = 25 m/s * 4 sec + ½ * 3 m/s2 * (4 sec)2 = 124 meters

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/s2, how long is the runway?

s = ut + ½at2
s = 20 m/s * 8 sec + ½ * 10 m/s2 * (8 sec)2 = 600 meters

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