Concept explainers
A ball is thrown with an initial speed υi at an angle θi with the horizontal. The horizontal range of the ball is R. and the ball reaches a maximum height R/6. In terms of R and g, find (a) the time interval during which the ball is in motion, (b) the ball’s speed at the peak of its path, (c) the initial vertical component of its velocity, (d) its initial speed, and (e) the angle θi, (f) Suppose the ball is thrown at the same initial speed found in (d) but at the angle appropriate for reaching the greatest height that it can. Find this height. (g) Suppose the ball is thrown at the same initial speed but at the angle for greatest possible range. Find this maximum horizontal range.
(a)
The time interval during which the ball is in motion.
Answer to Problem 4.56AP
The time interval during which the ball is in motion is
Explanation of Solution
Given info: The initial speed of the ball is
The motion of the ball follows the parabolic path and the ball is said to projectile, the motion of the ball is shown in the Figure below.
Figure (1)
The formula to calculate the maximum height reached by the projectile is,
Here,
Rearrange the above equation.
Substitute
Thus, the vertical component of the initial velocity is
The formula to calculate the time taken by the ball to reach the ground is,
Here,
Substitute
Conclusion:
Therefore, the time interval during which the ball is in motion is
(b)
The speed of the ball at the peak of its path.
Answer to Problem 4.56AP
The speed of the ball at the peak of its path is
Explanation of Solution
Given info: The initial speed of the ball is
From part (a) the time of flight is
From the Figure (1) the range of the ball and time of flight is,
Rearrange the above equation.
Substitute
Conclusion:
Therefore, the speed of the ball at the peak of its path is
(c)
The initial vertical component of the velocity.
Answer to Problem 4.56AP
The initial vertical component of the velocity is
Explanation of Solution
Given info: The initial speed of the ball is
From part (a) vertical component of the initial velocity is
Conclusion:
Therefore, the initial vertical component of the velocity is
(d)
The initial speed of the ball.
Answer to Problem 4.56AP
The initial velocity of the ball is
Explanation of Solution
Given info: The initial speed of the ball is
From part (a) vertical component of the initial velocity is,
Square both side of the above equation.
And from part (b) the horizontal component of the velocity is
Square both side of the above equation.
Add equation (1) and (2) to find the initial velocity.
Conclusion:
Therefore, the initial velocity of the ball is
(e)
The angle
Answer to Problem 4.56AP
The angle
Explanation of Solution
Given info: The initial speed of the ball is
From part (a) vertical component of the initial velocity is,
And from part (b) the horizontal component of the velocity is
Take the ratio of the horizontal component and the vertical component of the initial velocity.
Conclusion:
Therefore, the angle
(f)
The maximum height that the ball can reach with the same initial velocity.
Answer to Problem 4.56AP
The maximum height that the ball can reach with the same initial velocity is
Explanation of Solution
Given info: The initial speed of the ball is
For the maximum height to be gained by the ball the angle made by the horizontal should be
The formula to calculate the maximum height reached by the projectile is,
Here,
Rearrange the above equation.
From part (d) the initial velocity of the ball is,
Substitute
Conclusion:
Therefore, the maximum height that the ball can reach with the same initial velocity is
(g)
The maximum range of the ball with the same initial velocity.
Answer to Problem 4.56AP
The maximum range of the ball with the same initial velocity is
Explanation of Solution
Given info: The initial speed of the ball is
For the maximum range to be gained by the ball the angle made by the horizontal should be
The formula to calculate the maximum height reached by the projectile is,
Here,
From part (d) the initial velocity of the ball is,
Substitute
Conclusion:
Therefore, the maximum range of the ball with the same initial velocity is
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