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Motion (kinematics)

Velocity and displacement: v² = u² + 2as

Links speeds and distance under constant acceleration, without needing the time.

v2=u2+2as

Calculator

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

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)
s
the displacement in metres: distance in a chosen positive direction (it can be negative) (m)

When to use it

Use it when time is neither given nor asked for: the speed of an object after falling a height, the stopping distance of a car from its speed and braking deceleration. Solving for v gives two signs; + if the object is still moving in the positive direction, − if it has turned round.

Worked example

An object falls 20 m from rest (u = 0, a = 9.81 m/s², s = 20). How fast is it moving at the bottom?

  1. Start from the formula.

    v2=u2+2⁢a⁢s

  2. Rearrange to make v the subject: take the square root of both sides. A square root has two signs: + if the object is still moving in the positive direction, − if it is coming back.

    v=u2+2⁢a⁢s or −u2+2⁢a⁢s

  3. Moving forwards (+): substitute u = 0, a = 9.81, s = 20 and work it out.

    v=02+2·9.81·20
    =0+19.62·20
    =0+392.4
    =392.4
    ≈19.8091m/s

  4. Moving backwards (−): substitute u = 0, a = 9.81, s = 20 and work it out.

    v=−02+2·9.81·20
    =−0+19.62·20
    =−0+392.4
    =−392.4
    ≈−19.8091m/s

Answer v ≈ 19.81 m/s downwards (the other root, −19.81, would mean moving the opposite way)

Why it works

Eliminate t between v=u+at and s=12(u+v)t: from the first, t=(v−u)/a; substituting gives s=(v+u)(v−u)2a=v2−u22a.

Common mistakes

  • Taking a square root of a negative number: if u2+2as<0 the object never gets that far.
  • Forgetting to square u, or taking the square root of only part of the right side.
  • Using a positive a for braking; deceleration in the direction of motion is negative.