BIO Biomechanics. The mass of a regulation tennisball is 57 g (although it can vary slightly), and tests have shownthat the ball is in contact with the tennis racket for 30ms. (Thisnumber can also vary, depending on the racket and swing.) We shall assume a 30.0ms contact time. The fastest-known servedtennis ball was served by “Big Bill” Tilden in 1931, and its speed was measured to be 73m/s. (a) What impulse and what force didBig Bill exert on the tennis ball in his record serve? (b) If Big

Bill’s opponent returned his serve with a speed of 55m/s, whatforce and what impulse did he exert on the ball, assuming onlyhorizontal motion?

Given in the question

Mass of the tennis ball$m=57\mathrm{g}=57\times {10}^{-3}\mathrm{kg}$

Velocity of the served ball$v_2=73.14\mathrm{m}/\mathrm{s}$

Time of contact$∆t=30\mathrm{ms}$

The momentum of a particle: The momentum $\overrightarrow{\mathbf{P}}$of a particle is a vector quantity equal to the product ofthe particle’s mass mand velocity $\overrightarrow{\mathbf{v}}$.

$\overrightarrow{\mathbf{p}}\mathbf{=}\mathit{m}\overrightarrow{\mathbf{v}}$

Newton’ssecond law says that the net force on a particleis equal to the rate of change of the particle’smomentum.

$\mathbf{\sum }_{\mathbf{}}\mathit{F}\mathbf{=}\frac{\mathbf{d}\mathbf{p}}{\mathbf{d}\mathbf{t}}$

Impulse and momentum theorem

If $\overrightarrow{\mathbf{J}}$is impulse and $\overrightarrow{{\mathbf{P}}_{\mathbf{1}}}$is initial momentum and$\overrightarrow{{\mathbf{P}}_{\mathbf{2}}}$ is final momentum

Then

$\overrightarrow{\mathbf{J}}\mathbf{=}{\overrightarrow{\mathbf{P}}}_{\mathbf{2}}\mathbf{-}\overrightarrow{{\mathbf{P}}_{\mathbf{1}}}$

Given in the question

$$\begin{gathered} m=57\mathrm{g}=57\times {10}^{-3}\mathrm{kg} \\ {\mathrm{v}}_2=73.14\mathrm{m}/\mathrm{s} \\ ∆\mathrm{t}=30\mathrm{ms} \end{gathered}$$

As we know that service means hitting the ball and starting the play, since the ball is initially at rest so initial momentum is zero.

$p_1=0$

Final momentum

$$\begin{gathered} p_2=m{v}_2 \\ =\left(57\times {10}^{-3}\mathrm{kg}\right)\left(73.14\mathrm{m}/\mathrm{s}\right) \\ =4.17\mathrm{kg}.\mathrm{m}/\mathrm{s} \end{gathered}$$

From impulse momentum theorem.

$$\begin{gathered} \overrightarrow{J}=\overrightarrow{P_2}-\overrightarrow{P_1} \\ =4.17\mathrm{kg}.\mathrm{m}/\mathrm{s}-0 \\ =4.17\mathrm{kg}.\mathrm{m}/\mathrm{s} \end{gathered}$$

The impulse exerted by Big Bill on the tennis ball in his record serve is 4.17kg.m/s

From Newton’s second law

$$\begin{gathered} \sum _{}F=\frac{dp}{dt} \\ \sum _{}F=\frac{p_2-p_1}{∆t} \\ =\frac{4.17\mathrm{kg}.\mathrm{m}/\mathrm{s}-0}{30\times {10}^{-3}\mathrm{s}} \\ =138.97\mathrm{N} \end{gathered}$$

The force exerted by Big Bill on the tennis ball in his record serve is 138.97 N.

Given in the question

$m=57\mathrm{g}=57\times {10}^{-3}\mathrm{kg}$

Considering the direction from bill to his opponent is a positive direction.

$$\begin{gathered} {\mathrm{v}}_2=-55.0\mathrm{m}/\mathrm{s} \\ ∆\mathrm{t}=30\mathrm{ms} \end{gathered}$$

After Big Bill serves the ball momentum of the ball is

$p_1=4.17\mathrm{kg}.\mathrm{m}/\mathrm{s}$

Final momentum

$$\begin{gathered} p_2={\mathrm{mv}}_2 \\ =\left(57\times {10}^{-3}\mathrm{kg}\right)\left(-55\mathrm{m}/\mathrm{s}\right) \\ =-3.135\mathrm{kg}.\mathrm{m}/\mathrm{s} \end{gathered}$$

From impulse momentum theorem.

$$\begin{gathered} \overrightarrow{J}=\overrightarrow{P_2}-\overrightarrow{P_1} \\ =-3.135\mathrm{kg}.\mathrm{m}/\mathrm{s}-4.17\mathrm{kg}.\mathrm{m}/\mathrm{s} \\ =-7.3\mathrm{kg}.\mathrm{m}/\mathrm{s} \end{gathered}$$

The impulse exerted by the opponent is -7.3kg.m/s

From Newton’s second law

$$\begin{gathered} \sum _{}F=\frac{dp}{dt} \\ \sum _{}F=\frac{{\mathrm{p}}_2-{\mathrm{p}}_1}{∆\mathrm{t}} \\ =\frac{-3.135\mathrm{kg}.\mathrm{m}/\mathrm{s}-4.17\mathrm{kg}.\mathrm{m}/\mathrm{s}}{30\times {10}^{-3}\mathrm{s}} \\ =-243.5\mathrm{N} \end{gathered}$$

The force exerted by the opponents -243.5 N.