A ball is thrown up onto a roof, landing 4.16 s later at height h = 15.0 m above

Question

A ball is thrown up onto a roof, landing 4.16 s later at height h = 15.0 m above the release level. The ball's path just before landing is angled at = 60.0° with the roof. (Let the positive x direction be to the right, and the positive y direction be upward.)

a) Find the horizontal distance d it travels. (Hint: One way is to reverse the motion, as if on video.)

b) What is the magnitude of the ball's initial velocity?

c) What is the angle (relative to the horizontal) of the ball's initial velocity? (counterclockwise from the +x axis)

Explanation / Answer

given
time = 4.16s
h = 15m
= 60°
V = initial velocity of the ball
H = maximum height reached by the ball
t = time to reach max height
T = time the ball reaches the roof = 4.16
T - t = time it takes the ball to fall from max height to the roof
H - 15 = distance the ball falls from max height to the roof
g = acceleration of gravity = 9.81 m/s^2

Use the equations for calculating max height and for the ball to fall from max height to the roof.
H = (1/2)*g*t^2
H - 15 = (1/2*g*(T - t)^2 = (1/2*)g*(4.16 - t)^2

Put the first equation into the second for H.
(1/2)*g*t^2 - 15 = (1/2)*g*(4.16 - t)^2 = (1/2)*g*[17.30 - 8.32*t + t^2]

t = 2.44 seconds

Now use this time to get the initial vertical velocity.
v = g*t = 9.81* 2.44= 23.93 m/s

Use the vertical impact velocity and angle to get the horizontal component of velocity at impact. This will equal the initial horizontal velocity since there are no forces acting in the horizontal.
Time to fall = 4.16 - 2.44 = 1.72 seconds
Vertical velocity = v = g*(time to fall) = 9.81*1.72 = 16.856 m/s

tan(60) = (vertical velocity)/(horizontal velocity)
Horizontal velocity = 16.856/tan(60) = 9.73m/s

The initial velocity is then:
V = SQRT[(9.73)^2 + (23.93)^2]
V = 25.83 m/s
Angle is:
tan(Angle) = (vertical velocity)/(horizontal velocity) = 23.93/9.73

Angle =67.87 degrees

Distance is just the horizontal speed times the time of flight.
Distance = 9.73*4.16
Distance = 40.55 meters

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