Concept explainers
(a)
The time taken by a ball released from an airplane to hit the ground.
Answer to Problem 69QAP
The ball hits the ground after 9.24 s.
Explanation of Solution
Given:
The speed of the airplane when the ball is released
Angle made by the plane to the horizontal
Height of the airplane above the ground when the ball is released
Formula used:
The time of flight of the ball is determined using the equation for the vertical motion of the ball.
Here,
Calculation:
When the airplane releases the ball, the ball has the velocity of the airplane. Its speed is 35.3 m/s and it is released at an angle 30.0o to the horizontal.
Assume the origin to be located at the point where the ball is released. With the x axis parallel to the ground and the + y axis directed upwards.
The ball travels a parabolic path and lands on the ground at point B. Its vertical displacement when it lands on the ground, is equal to
This is shown in the diagram below.
Calculate the vertical component of the ball's velocity.
The ball falls under the action of the gravitational force. Hence the acceleration acting on the ball in the vertical direction is the acceleration of free fall.
In equation (1), substitute
Solve the quadratic equation to determine t.
Taking the positive root,
Conclusion:
Thus, the ball hits the ground after 9.24 s.
(b)
The maximum height of the ball from the ground.
Answer to Problem 69QAP
The ball reaches a maximum height of 270.9 m from the ground.
Explanation of Solution
Given:
The speed of the airplane when the ball is released
Angle made by the plane to the horizontal
Height of the airplane above the ground when the ball is released
Formula used:
The maximum height reached by the ball can be calculated using the equation of motion,
Here, vy is vertical component of the ball's velocity at the position y,
The maximum height h reached by the ball, when measured from the ground is given by,
Calculation:
The vertical component of the ball's velocity reduces as it moves up, due to the action of the gravitational force. When the vertical component reaches a value zero, the ball can no longer make an upward displacement, hence after this point it starts its motion in the downward direction.
Therefore, at maximum height,
In equation (2) substitute 0 m/s for vy,
This point is 15.89 m above the point of projection. Therefore, its height from the ground is given by,
Conclusion:
Thus, the ball reaches a maximum height of 270.9 m from the ground.
(c)
The horizontal distance traveled by the ball from the point of release to the ground.
Answer to Problem 69QAP
The ball travels a horizontal distance of 282.5 m.
Explanation of Solution
Given:
The speed of the airplane when the ball is released
Angle made by the plane to the horizontal
Height of the airplane above the ground when the ball is released
Time of flight of the ball
Formula used:
The horizontal distance traveled by the ball is calculated using the equation
Here,
Calculation:
The ball makes a vertical displacement of
Calculate the horizontal component of the ball's velocity.
Substitute the values of v0x, ax and t in equation (3) and solve for
Conclusion:
Thus, the ball travels a horizontal distance of 282.5 m.
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