Chapter 36, Problem 1CQ

(a)

To determine

The speed and direction of each ball in the reference frames.

A reference frame that moves with the ball 1.

Answer to Problem 1CQ

Solution:

Ball 1 has speed 0 m/s appears to be at rest.

Ball 2 has speed 9 m/s and direction to the left.

Explanation of Solution

Given information :

The following figure if two balls are given

  

Calculation:

The relative velocities of the balls are used to find the speed and direction of the two balls with respect to the reference frames moving with the balls.

The direction to the right is taken as positive, and the left as negative.

The two balls are shown in the figure:

The ball 1 moves to the right, hence the velocity ( $u_1$ ) is taken as positive.

  $u_1=\text{ 6 m}/\text{s}$

The ball 2 moves to the left, hence the velocity ( $u_2$ ) is taken as negative.

  $u_2=-3\text{ m}/\text{s}$ .

A reference frame that moves with ball 1.

The velocity of the reference frame, $v_1=\text{ 6 m}/\text{s}$

The relative velocity of ball 1 ( $u_1{}^{\prime }$ ) with respect to the reference frame can be found as:

  $\begin{array}{c}{u}_1{}^{\prime }=u_1-v_1\\ =6\text{ m/s }-6\text{ m/s}\\ \text{= 0 m/s}\end{array}$

The relative velocity of ball 2 ( $u_2{}^{\prime }$ ) with respect to reference frame can be found as:

  $\begin{array}{c}{u}_2{}^{\prime }=u_2-v_1\\ =-3\text{ m/s }-6\text{ m/s}\\ \text{= -9 m/s}\end{array}$

Negative sign indicates direction to the left.

When the reference frame moves with the ball 1:

• Ball 1 has speed = 0 m/s and appears to be at rest.
• Ball 2 has speed = 9 m/s and has direction to the left.

Conclusion:

When the reference frame moves with the ball 1.

Ball 1 has speed = 0 m/s and appears to be at rest.

Ball 2 has speed = 9 m/s, direction to the left.

(b)

To determine

The speed and direction of each ball in the reference frames.

A reference frame that moves with the ball 2.

Answer to Problem 1CQ

Solution:

Ball 1 has speed of 9 m/s and direction to the right

Ball 2 has speed of 0 m/s and appears to be at rest.

Explanation of Solution

A reference frame that moves with the ball 2

The velocity of the reference frame, $v_2=-\text{3 m}/\text{s}$

The relative velocity of ball 1 ( $u_1{}^{\prime }$ ) with respect to reference frame can be found as :

  $\begin{array}{c}{u}_1{}^{\prime }=u_1-v_2\\ =6\text{ m/s }-(-3\text{ m/s)}\\ \text{= }6\text{ m/s }+3\text{ m/s}\\ \text{= 9 m/s}\end{array}$

Positive sign indicates direction to the right.

The relative velocity of ball 2 ( $u_2{}^{\prime }$ ) with respect to reference frame can be found as

  $\begin{array}{c}{u}_2{}^{\prime }=u_2-v_2\\ =-3\text{ m/s }-(-3\text{ m/s)}\\ \text{=}-3\text{ m/s }+3\text{ m/s}\\ \text{= }0\text{ m/s}\end{array}$

When the reference frame moves with ball 2.

• Ball 1 has speed = 9 m/s and has direction to the right
• Ball 2 has speed = 0 m/s and appears to be at rest.

Conclusion:

When the reference frame moves with ball 2.

Ball 1 has speed = 9 m/s and has direction to the right.

Ball 2 has speed = 0 m/s and appears to be at rest.