Concept explainers
A ball is thrown with an initial speed vi 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)
Theflight time of the ball in the motion .
Answer to Problem 54P
The flight time of the ball in the motion is
Explanation of Solution
Write the expression for the maximum height of the ball,
Here,
Write the expression for the horizontal range of the ball,
Here,
Substitute
Combine (I) and (II) and substitute
Rewrite the relation for
Write the expression for the vertical velocity of the ball,
Here,
Conclusion:
Substitute
Total time of the ball’s flight,
Therefore, theflight time of the ball in the motion is
(b)
Thespeed of the ball at the peak of its path.
Answer to Problem 54P
The speed of the ball at the peak of its path is
Explanation of Solution
Write the expression for the speed of the ball at the path’s peak,
Here,
Conclusion:
Substitute
Therefore, thespeed of the ball at the peak of its path is
(c)
The initial vertical component of the ball’s velocity .
Answer to Problem 54P
The initial vertical component of the ball’s velocity is
Explanation of Solution
Write the expression for the initial vertical velocity of the ball,
Here,
Conclusion:
Substitute
Therefore, theinitial vertical component of the ball’s velocity is
(d)
The initial speed of the ball.
Answer to Problem 54P
The initial speed of the ball is
Explanation of Solution
Write the expression for the initial speed of the ball,
Here,
Rewrite the above expression,
Conclusion:
Substitute
Therefore, theinitial speed of the ball is
(e)
The projectile angle of the ball .
Answer to Problem 54P
The projectile angle of the ballis
Explanation of Solution
Equation (III) divided by (V),
Conclusion:
Rewrite the above equation,
Therefore, theprojectile angle of the ball is
(f)
The maximum height of the ball when it throws at maximum projection angle.
Answer to Problem 54P
The maximum height of the ball when it throws at maximum projection angle is
Explanation of Solution
In this case, the maximum projection angle must be
From (I),
Write the expression for the maximum height of the ball thrown,
Here,
Conclusion:
Substitute
Therefore, themaximum height of the ball when it throws at maximum projection angle is
(g)
The maximum horizontal range of the ball.
Answer to Problem 54P
The maximum horizontal range of the ballis
Explanation of Solution
In this case, the maximum projection angle must be
Write the expression for the maximum range of the ball thrown,
Here,
Conclusion:
Substitute
Therefore, themaximum horizontal range of the ball is
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Chapter 3 Solutions
Principles of Physics
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