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Motion (kinematics)

Final velocity: v = u + at

The velocity after accelerating steadily at a for time t, starting from velocity u.

v=u+at

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−0.20.20.40.60.811.21.41.61.822.25101520uv
  • Slope = acceleration a = 9.81 m/s²; final velocity v = 19.62 m/s.
  • Shaded area = displacement s = ut + ½at² = 19.62 m (area below the axis counts as negative).

What the symbols mean

v
the final velocity in m/s (m/s)
u
the initial velocity in m/s (m/s)
a
the acceleration in m/s², assumed constant; negative if it acts against the positive direction (g = 9.81 m/s² for free fall) (m/s²)
t
the time in seconds (s)

When to use it

Use it when acceleration is constant and the problem involves velocities and time but not distance: a car speeding up, a ball thrown upwards (a = −9.81 m/s² if up is positive), anything dropped from rest (u = 0). It is the SUVAT equation without s.

Worked example

A stone is dropped from rest (u = 0) and falls with a = 9.81 m/s². How fast is it falling after t = 2 s?

  1. Start from the formula.

    v=u+a⁢t

  2. Substitute u = 0, a = 9.81, t = 2 and work it out.

    v=0+9.81·2
    =0+19.62
    =19.62m/s

Answer v = 19.62 m/s

Why it works

Constant acceleration means the velocity changes by a every second, so after t seconds it has changed by at. On a velocity–time graph it is a straight line with slope a.

Common mistakes

  • Giving a deceleration a positive sign; if the object slows down in the positive direction, a is negative.
  • Using it when acceleration is not constant (for example with air resistance at high speed).
  • Mixing km/h with m/s²; convert speeds to m/s first.