Concept explainers
(a)
The total flight time of the ball.
(a)
Answer to Problem 56AP
The total fight time of the ball is
Explanation of Solution
The initial velocity is divided into two components that is horizontal and vertical velocity component. At maximum point vertical velocity is zero for this path. The maximum height of the projectile is sixth part of the range.
Write the expression for maximum height of the ball.
Here,
Write the expression for range of the projectile path.
Substitute
Here,
Multiply equation (II) by
Multiply and divide by
The maximum height is one-sixth of range.
Substitute
Simplify the above equation.
Substitute
Simplify and rearrange the above equation as.
At peak point of the projectile path velocity along vertical direction is zero.
Write the expression for final velocity along vertical.
Here,
Initial velocity along vertical is zero.
Substitute
Rearrange the above equation.
The final velocity along vertical is
Substitute
Rearrange the above equation time to reach the peak of the path.
Here,
Substitute
Rearrange the above equation.
Write the expression for the total time of the ball’s flight.
Conclusion:
Substitute
Thus, the total flight time of the ball is
(b)
The speed of the ball at peak point.
(b)
Answer to Problem 56AP
The peak velocity at top point is
Explanation of Solution
At the peak point velocity along vertical is zero and the ball starts to accelerate under gravity in downward direction.
Write the expression for velocity at peak point.
Here,
Conclusion:
Substitute
Thus, the peak velocity at top point is
(c)
The initial vertical component of velocity.
(c)
Answer to Problem 56AP
The initial vertical component is
Explanation of Solution
Write the expression for initial vertical component of velocity.
Here,
Conclusion:
Substitute
Thus, the initial vertical component of velocity is
(d)
The initial speed of the ball.
(d)
Answer to Problem 56AP
The initial speed of the ball is
Explanation of Solution
The initial speed of the ball is given by the square root of the sum of square of horizontal and vertical velocity component.
Write the expression for initial velocity along horizontal.
Here,
Write the expression for initial velocity along vertical.
Here,
Write the expression for the initial speed of ball.
Here,
Conclusion:
Substitute
Substitute
Simplify and solve the above equation.
Thus, the initial speed of the ball is
(e)
The initial angle of the projectile.
(e)
Answer to Problem 56AP
The initial angle of projectile is
Explanation of Solution
The angle of the projectile can be given by the ratio of vertical velocity to the horizontal velocity.
Write the expression for initial angle of projectile as.
Substitute
Here,
Conclusion:
Substitute
Thus, the initial projection angle is
(f)
The greatest height with same initial speed.
(f)
Answer to Problem 56AP
The greatest height at same initial speed is
Explanation of Solution
The maximum height can be achieved by the ball at
Conclusion:
Substitute
Thus, the greatest height at same initial speed is
(g)
The maximum horizontal range.
(g)
Answer to Problem 56AP
The maximum horizontal range is
Explanation of Solution
The maximum horizontal range can be achieved at
Write the expression for maximum range as.
Here,
Conclusion:
Substitute
Thus, the maximum horizontal range is
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Chapter 4 Solutions
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
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