Concept explainers
(a)
The total time of flight of the ball as observed by the juggler in the train.
(a)
Answer to Problem 64P
The total time of flight of the ball as observed by the juggler in the train is found to be
Explanation of Solution
Given:
The initial velocity of the ball relative to the train
Formula used:
To determine the time of flight
Here,
Calculation:
The ball thrown in a train and the juggler are at rest with respect to the train. Choose a one dimensional coordinate system with the origin on the train and the positive y axis directed upwards. As the ball moves up and returns to the juggler’s hands, its displacement
Therefore,
Substitute these values in equation (1) and calculate the time of flight.
Conclusion:
Thus, the total time of flight of the ball as observed by the juggler in the train is found to be
(b)
The displacement of the ball during its rise as observed by the juggler.
(b)
Answer to Problem 64P
The displacement of the ball during its rise as observed by the juggler, is found to be
Explanation of Solution
Given:
The initial velocity of the ball relative to the train
Formula used:
To determine the displacement of the ball , the following equation of motion may be used.
Calculation:
As the ball moves upwards, it slows down due to the action of the Earth’s gravitational force. At the top most point of its trajectory, is instantaneous velocity v becomes zero.
Substitute the values of variables in the equation (2) and solve for
Conclusion:
Thus, the displacement of the ball during its rise as observed by the juggler, is found to be
(c)
The ball’s initial speed as observed by the friend on the ground.
(c)
Answer to Problem 64P
The ball’s initial speed as observed by the friend on the ground is found to be
Explanation of Solution
Given:
The initial velocity of the ball relative to the train
The velocity of the train relative to the ground
Formula used:
Using a coordinate system with the origin at the ground and the positive x axis along East, a vector diagram is constructed.
Figure 1
The person on the ground observes the ball to have a velocity
Calculation:
Substitute the values of variables in equation (3) and calculate the speed of the ball as observed by the person on the ground.
Conclusion:
Thus, the ball’s initial speed as observed by the friend on the ground is found to be
(d)
The angle of launch of the ball as observed by the person on the ground.
(d)
Answer to Problem 64P
The angle of launch of the ball as observed by the person on the ground
Explanation of Solution
Given:
The initial velocity of the ball relative to the train
The velocity of the train relative to the ground
Formula used:
Use Figure 1 to calculate the angle
Calculation:
Substitute the values of the variables in equation (4) and calculate the angle of launch of the ball as observed by the person on the ground.
Conclusion:
Thus, the angle of launch of the ball as observed by the person on the ground
(e)
The displacement of the ball during its rise as observed by the person on the ground.
(e)
Answer to Problem 64P
The displacement of the ball during its rise as observed by the person on the ground is found to be
Explanation of Solution
Given:
The initial velocity of the ball relative to the train
The velocity of the train relative to the ground
Formula used:
The displacement of the ball as seen by the person on the ground is given by the expression,
Here,
From Figure 1, it can be seen that the initial velocity
The time
The horizontal component of the ball’s velocity remains constant, since no force acts along the horizontal direction. While, since the acceleration of free fall acts downwards, the vertical component of the ball’s velocity varies with time.
The values of
Calculation:
The trajectory of the ball as seen by the person on the ground is shown in the diagram below;
At the top most point of its trajectory, the vertical component of the ball’s velocity becomes equal to zero. Use equation (7) and calculate the time taken by the ball to reach the top most point of its trajectory.
Calculate the value of
Calculate the value of
Substitute the values of
Conclusion:
Thus, the displacement of the ball during its rise as observed by the person on the ground is found to be
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Chapter 3 Solutions
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