Question

A 57g tennis ball is served at 45m/s .

If the ball started from rest, what impulse was applied to the ball by the racket?

Express your answer using two significant figures.

Answer 1

Concepts and reason

The concepts used to solve this problem are the momentum expression and change in momentum principle.

First calculate the momentum of the ball by using the momentum formula. After that, calculate the impulse by using change in momentum principle.

Fundamentals

The expression of momentum is,

$P = mv$

Here, $m$ is mass, and $v$ is velocity.

The impulse is equal to the change in momentum.

$\begin{array}{c}\\I = \Delta P\\\\ = {P_{\rm{f}}} - {P_{\rm{i}}}\\\end{array}$

Here, ${P_{\rm{f}}}$ is final momentum, and ${P_{\rm{i}}}$ is initial momentum.

Calculate the final momentum of the ball.

The expression of momentum is,

$P = mv$

Here, $m$ is mass, and $v$ is velocity.

Substitute $57{\rm{ g}}$ for $m$ , and $45{\rm{ m/s}}$ for $v$ .

$\begin{array}{c}\\{P_{\rm{f}}} = \left( {57{\rm{ g}}} \right)\left( {45{\rm{ m/s}}} \right)\\\\ = \left( {57{\rm{ g}}} \right)\left( {\frac{{{{10}^{ - 3}}{\rm{ kg}}}}{{1{\rm{ g}}}}} \right)\left( {45{\rm{ m/s}}} \right)\\\\ = \left( {57 \times {{10}^{ - 3}}{\rm{ kg}}} \right)\left( {45{\rm{ m/s}}} \right)\\\\ = 2.6{\rm{ kg}} \cdot {\rm{m/s}}\\\end{array}$

Calculate the in initial momentum.

Because the ball is initially in rest so the initial momentum of the ball is zero.

${P_{\rm{i}}} = 0$

Calculate the impulse applied to the ball by the rocket.

The impulse is equal to the change in momentum.

$\begin{array}{c}\\I = \Delta P\\\\ = {P_{\rm{f}}} - {P_{\rm{i}}}\\\end{array}$

Here, ${P_{\rm{f}}}$ is final momentum, and ${P_{\rm{i}}}$ is initial momentum.

Substitute $2.6{\rm{ kg}} \cdot {\rm{m/s}}$ for ${P_{\rm{f}}}$ , and $0\;{\rm{kg}} \cdot {\rm{m/s}}$ for ${P_{\rm{i}}}$ .

$\begin{array}{c}\\I = \left( {2.6{\rm{ kg}} \cdot {\rm{m/s}}} \right) - \left( {0\;{\rm{kg}} \cdot {\rm{m/s}}} \right)\\\\ = 2.6{\rm{ kg}} \cdot {\rm{m/s}}\\\end{array}$

Ans:

The impulse applied to the ball by the rocket $2.6{\rm{ kg}} \cdot {\rm{m/s}}$ .