Impulse equals the change in momentum

Newton's Laws & Momentum — AQA A-Level Physics

Key Definition

Impulse — The product of force and the time for which it acts. Equal to the change in momentum. Unit: N s (equivalent to kg m s^-1).

$$F\Delta t = \Delta p$$

\begin{aligned} F \Delta t &= \Delta p \\ &= mv - mu \end{aligned}

$F$: resultant force (N)
$Δt$: time interval (s)
$Δp$: change in momentum (kg m s⁻¹)
$m$: mass (kg)
$v$: final velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹. (m s⁻¹)
$u$: initial velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹. (m s⁻¹)

ImpulseThe product of force and the time for which it acts. Equal to the change in momentum. is a vector -- direction matters.
A large force over a short time gives the same impulseThe product of force and the time for which it acts. Equal to the change in momentum. as a small force over a long time.
On a force-time graph, the impulseThe product of force and the time for which it acts. Equal to the change in momentum. is the area under the curve.
If the force varies, count squares or split into geometric shapes to find the area.

Worked Example

A 58 g tennis ball moving left at 30 m s^-1 is struck and returns to the right at 20 m s^-1. Find the impulse.

Show Solution

Define positive direction and list values

Taking left as positive:

$u = 30$ m s$^{-1}$
$v = -20$ m s$^{-1}$ (returns right)
$m = 0.058$ kg

Calculate impulse

$$\text{Impulse} = m(v - u) = 0.058 \times (-20 - 30) = 0.058 \times (-50) = -2.9 \text{ N s}$$

State direction

The negative sign means the impulse acts to the right (opposite to the initial motion).

Answer

$Impulse = 2.9 N s to the right.$

Common Mistake MEDIUM

Students often: Forgetting to change the sign of velocity when an object rebounds.
Instead: If an object changes direction, its final velocity must have the opposite sign to its initial velocity. $Impulse = m(v - u) will$ then give a larger magnitude than if both velocities had the same sign.