Chapter 10, Problem 4PQ

A mother pushes her son in a stroller at a constant speed of 1.52 m/s. The boy tosses a 56.7-g tennis ball straight up at 1.75 m/s and catches it. The boy’s father sits on a bench and watches.

a. According to the mother, what are the ball’s initial and final momenta?

b. According to the father, what are the ball’s initial and final momenta?

c. According to the mother, is the ball’s momentum ever zero? If so, when? If not, why not?

d. According to the father, is the ball’s momentum ever zero? If so, when? If not, why not?

To determine

The initial and final momentum of ball according to mother’s view.

Answer to Problem 4PQ

The initial momentum is $\underset{\_}{9.92\times {10}^{-2}\widehat{j}\text{kg}\cdot \text{m}/\text{s}}$, and final momentum is $\underset{\_}{-9.92\times {10}^{-2}\widehat{j}\text{kg}\cdot \text{m}/\text{s}}$.

Explanation of Solution

Consider the figure showing the motion of ball on the basis of mother’s point of view.

According to mother’s point of view, the ball moves vertically upwards and falls vertically downwards with same speed. But initial momentum is in positive y direction and final momentum is in negative y direction.

Write the expression for initial momentum.

  ${\left(\overrightarrow{p_i}\right)}_{\text{mot}}=m\overrightarrow{v_i}$                                                                                                             (I)

Here, $m$ is the mass of ball, and $v_i$ is the initial velocity.

Write the expression for final momentum.

  ${\left(\overrightarrow{p_f}\right)}_{\text{mot}}=m\overrightarrow{v_f}$                                                                                               (II)

Here, $m$ is the mass of ball, and $v_f$ is the final velocity.

Conclusion:

Substitute, $56.7\times {10}^{-3}\text{kg}$ for $m$, and $1.75\widehat{j}\text{m/s}$ for $v_i$ in equation (I).

  $\begin{array}{c}{\left(\overrightarrow{p_i}\right)}_{\text{mot}}=\left(56.7\times {10}^{-3}\text{kg}\right)\left(1.75\widehat{j}\text{m/s}\right)\\ =9.92\times {10}^{-2}\widehat{j}\text{kg}\cdot \text{m/s}\end{array}$

Substitute, $56.7\times {10}^{-3}\text{kg}$ for $m$, and $-1.75\widehat{j}\text{m/s}$ for $v_f$ in equation (II).

  $\begin{array}{c}{\left(\overrightarrow{p_i}\right)}_{\text{mot}}=\left(56.7\times {10}^{-3}\text{kg}\right)\left(-1.75\widehat{j}\text{m/s}\right)\\ =-9.92\times {10}^{-2}\widehat{j}\text{kg}\cdot \text{m/s}\end{array}$

Therefore, the initial momentum is $\underset{\_}{9.92\times {10}^{-2}\widehat{j}\text{kg}\cdot \text{m}/\text{s}}$, and final momentum is $\underset{\_}{-9.92\times {10}^{-2}\widehat{j}\text{kg}\cdot \text{m}/\text{s}}$.

To determine

The initial and final momentum of ball according to father’s view.

Answer to Problem 4PQ

The initial momentum is $\underset{\_}{\left(8.62\times {10}^{-2}\widehat{i}+9.92\times {10}^{-2}\widehat{j}\right)\text{kg}\cdot \text{m}/\text{s}}$, and final momentum is $\underset{\_}{\left(8.62\times {10}^{-2}\widehat{i}-9.92\times {10}^{-2}\widehat{j}\right)\text{kg}\cdot \text{m}/\text{s}}$.

Explanation of Solution

Consider the figure showing the motion of ball on the basis of father’s point of view

The path of ball is now parabolic, horizontal velocity is same as those observed by mother, but vertical velocity is equal to the velocity of stroller.

Write the expression for initial momentum.

  ${\left(\overrightarrow{p_i}\right)}_{dad}=m\overrightarrow{v_i}$                                                                                                           (III)

Here, $m$ is the mass of ball, and $v_i$ is the initial velocity observed by dad.

Write the expression for final momentum.

  ${\left(\overrightarrow{p_f}\right)}_{\text{dad}}=m\overrightarrow{v_f}$                                                                                                        (IV)

Here, $m$ is the mass of ball, and $v_f$ is the final velocity observed by dad.

Conclusion:

Substitute, $56.7\times {10}^{-3}\text{kg}$ for $m$, and $\left(1.52\widehat{i}+1.75\widehat{j}\right)\text{m/s}$ for $v_i$ in equation (III).

  $\begin{array}{c}{\left(\overrightarrow{p_i}\right)}_{dad}=\left(56.7\times {10}^{-3}\text{kg}\right)\left(1.52\widehat{i}+1.75\widehat{j}\right)\text{m/s}\\ =\left(8.62\times {10}^{-2}\widehat{i}+9.92\times {10}^{-2}\widehat{j}\right)\text{kg}\cdot \text{m/s}\end{array}$

Substitute, $56.7\times {10}^{-3}\text{kg}$ for $m$, and $\left(1.52\widehat{i}-1.75\widehat{j}\right)\text{m/s}$ for $v_i$ in equation (IV).

  $\begin{array}{c}{\left(\overrightarrow{p_i}\right)}_{dad}=\left(56.7\times {10}^{-3}\text{kg}\right)\left(1.52\widehat{i}-1.75\widehat{j}\right)\text{m/s}\\ =\left(8.62\times {10}^{-2}\widehat{i}-9.92\times {10}^{-2}\widehat{j}\right)\text{kg}\cdot \text{m/s}\end{array}$

Therefore, the initial momentum is $\underset{\_}{\left(8.62\times {10}^{-2}\widehat{i}+9.92\times {10}^{-2}\widehat{j}\right)\text{kg}\cdot \text{m}/\text{s}}$, and final momentum is $\underset{\_}{\left(8.62\times {10}^{-2}\widehat{i}-9.92\times {10}^{-2}\widehat{j}\right)\text{kg}\cdot \text{m}/\text{s}}$.

To determine

Whether the momentum of ball is zero ever according to mother.

Answer to Problem 4PQ

The momentum will be zero according to mother’s point of view.

Explanation of Solution

According to mother’s view the ball is in vertical motion. At maximum height the kinetic energy is completely converted to potential energy, and velocity of ball is zero, or ball is momentarily at rest.

Momentum is the product of mass and velocity, thus, if velocity is zero, momentum will be zero, at top of flight.

Conclusion:

Therefore, momentum will be zero, at maximum height according to mother’s view.

To determine

Whether the momentum of ball is zero ever according to father.

Answer to Problem 4PQ

The momentum is never become zero according to father’s view.

Explanation of Solution

The motion is parabolic according to father’s view, and has two components of velocity, horizontal and vertical. Even though vertical component become zero, at any point, the horizontal component remains same until the ball covers entire parabolic path.

Hence momentum will not be equal to zero.

Conclusion:

The total velocity is the sum of horizontal and vertical velocity, since horizontal velocity remains same as a nonzero value, momentum is never become zero.