← Kinematic equations

Choosing the right equation

A car slows from 14 m/s to 5 m/s along a stretch of track. The report gives no times at all. Which kinematic equation can work with exactly the numbers given?

Here is a jet starting its takeoff run: 80 m/s already, accelerating at 4 m/s² along 600 m of runway.

How fast is it moving at the end? You now have four equations — and only one of them fits these numbers.

The steps are the same as always. 1. Write down the values in the question, and mark the unknown.

quantityvaluev?← find thisv₀80 m/sa4 m/s²Δx600 mt—

2. Select the correct equation. The jet's table gives v₀, a, and Δx, and asks for v — the time appears nowhere. So the equation you need is the one with no t in it: v² = v₀² + 2aΔx.

equationleaves outv = v₀ + atno Δx in itΔx = v₀t + ½at²no v in itv² = v₀² + 2aΔxno t in itΔx = ½(v₀ + v)tno a in it
That is the whole trick: the quantity missing from your table points at the one equation that leaves it out.

A boat moving at 6 m/s speeds up at 0.5 m/s² for 40 s. Which kinematic equation finds its final velocity?

Write down the values in the question:

v = ?

v₀ = 6 m/s

a = 0.5 m/s²

t = 40 s

Select the correct equation:

Δx appears nowhere — the equation you need is the one with no Δx in it.

v = v₀ + at

A cyclist moving at 4 m/s accelerates at 2 m/s² for 6 s. Which equation finds her velocity at the end of that time?

A driver slows from 21 m/s to 9 m/s over 90 m. Which kinematic equation finds the acceleration?

Write down the values in the question:

a = ?

v₀ = 21 m/s

v = 9 m/s

Δx = 90 m

Select the correct equation:

t appears nowhere, so the equation you need is the one with no t in it.

v² = v₀² + 2aΔx

A cart enters a ramp at 3 m/s and accelerates at 1.5 m/s² for 4 s. Which equation finds how far it travels in that time?

quantitiesgivenv₀, v, taskedΔxin neither listachooseΔx = ½(v₀ + v)t
No acceleration anywhere in the problem: choose Δx = ½(v₀ + v)t, the equation that leaves a out.

A jet moving at 80 m/s accelerates at 4 m/s² along 600 m of runway. Which equation finds its velocity at the end of that stretch?

Back to the jet: its problem gives 80 m/s, 4 m/s², and 600 m, and asks for the final velocity. The time appears nowhere — so the equation you need is v² = v₀² + 2aΔx, and you knew it before any algebra started.

Your turn

An elevator's velocity climbs steadily from 1 m/s to 5 m/s during a 4 s stretch. Which equation finds how far it rises in that stretch?

A driver slows from 24 m/s to 9 m/s over 45 m. Which equation finds the acceleration?

A bus rolling at 12 m/s brakes at −1.5 m/s² until it comes to a stop. Which equation finds how long the braking takes?