← Kinematic equations

The no-time equation: v² = v₀² + 2aΔx

A motorcycle moving at 8 m/s accelerates at a constant 3 m/s² for 4 s. Calculate its final velocity, in m/s.

Here is a van driving at a steady 7 m/s. The driver hits the accelerator and holds it for a 20 m stretch of road, accelerating at 3 m/s². How fast is the van moving at the end of the stretch?

in at 7 m/s, accelerating at 3 m/s²20 mout at v = ?
It enters the stretch at 7 m/s and speeds up the whole way. How fast is it moving 20 m later?

First, write down the values from the question, and mark the one you want.

quantityvaluev?← find thisv₀7 m/sΔx20 ma3 m/s²t—

Next, choose an equation. Previously, you've calculated a final velocity with v = v₀ + at — but it doesn't work here: we have v₀ and a, but not t.

Luckily, there's a third equation we can use.

v² = v₀² + 2aΔxthe no-time equation
symbolnameunitvfinal velocitym/sv₀starting velocitym/saaccelerationm/s²Δxdisplacementm

A rowing boat accelerates from 2 m/s to 5 m/s over 21 m. In v² = v₀² + 2aΔx, which of the boat's numbers plays the part of v₀?

No time in it — use it when the question gives a distance instead of a time.

Later, the driver hits the brakes: the van slows from 13 m/s to 5 m/s over 18 m. Write the five-quantity table for this stretch.

Write down the values in the question:

v = 5 m/s

v₀ = 13 m/s

Δx = 18 m

No time is given, and none is asked for.

The question doesn't give a. That's the unknown to find.

The table reads: v = 5 m/s, v₀ = 13 m/s, Δx = 18 m, a = ? (find this). No time is given.

Your turn

A cyclist speeds up from 4 m/s to 9 m/s down a slope of known length. No stopwatch was running. Which equation lets you find the cyclist's acceleration from exactly these numbers?

A boat slows from 8 m/s to 3 m/s while covering 44 m. In v² = v₀² + 2aΔx, which of the boat's numbers plays the part of Δx?

Four students write down the no-time kinematic equation from memory. Who has it right?