FHSST Physics/Momentum/Change

Change in Momentum

If either an object's mass or velocity changes then its momentum too will change. If an object has an initial velocity ${\overrightarrow {u}}$ and a final velocity ${\overrightarrow {v}}$, then its change in momentum, $\Delta {\overrightarrow {p}}$, is

$\Delta {\overrightarrow {p}}$

= ${\overrightarrow {p}}_{final}-{\overrightarrow {p}}_{initial}$ $m{\overrightarrow {v}}-m{\overrightarrow {u}}$

Worked Example 35 Change in Momentum

Question: A rubber ball of mass 0.8kg is dropped and strikes the floor at a velocity of $6\ m.s^{-1}$. It bounces back with an initial velocity of $4\ m.s^{-1}$. Calculate the change in momentum of the rubber ball caused by the floor.

Answer:

Step 1 :

Analyse the question to determine what is given. The question explicitly gives

• the ball's mass,
• the ball's initial velocity, and
• the ball's final velocity

all in the correct units.

Do not be confused by the question referring to the ball bouncing back with an ``initial velocity of $4\ m.s^{-1}$. The word initial is included here since the ball will obviously slow down with time and $4\ m.s^{-1}$ is the speed immediately after bouncing from the floor.

Step 2 :

What is being asked? We are asked to calculate the change in momentum of the ball,

${\begin{matrix}\Delta {\overrightarrow {p}}&=&m{\overrightarrow {v}}-m{\overrightarrow {u}}.\end{matrix}}$

We have everything we need to find $\Delta {\overrightarrow {p}}$. Since the initial momentum is directed downwards and the final momentum is in the upward direction, we can use the algebraic method of subtraction discussed in the vectors chapter.

Step 3 : Firstly, we choose a positive direction. Let us choose down as the positive direction. Then substituting,

Down is the positive direction

${\begin{matrix}\Delta {\overrightarrow {p}}&=&m{\overrightarrow {v}}-m{\overrightarrow {u}}\\&=&(0.8kg)(-4\ m.s^{-1})-(0.8kg)(+6\ m.s^{-1})\\&=&(0.8kg)(-10\ m.s^{-1})\\&=&-8\ kg.m.s^{-1}\\&=&8\ kg.m.s^{-1}{\textbf {\ up}}\end{matrix}}$

where we remembered in the last step to include the direction of the change in momentum in words.