(Figure 1) shows a friend standing on the roof of a building that is 51.8 m tall

Question

(Figure 1) shows a friend standing on the roof of a building that is 51.8 m tall. The roof is square and measures 20 m on a side. You want to shoot a paintball so that it lands on the roof and startles your friend, using a gun that shoots paintballs at a muzzle speed of 42 m/s. The only problem is a slim billboard 67.5 mhigh between you and the roof, 20 m in front of the building. You position yourself in front of the billboard such that when you hold the gun 1.5 m above the ground and fire, the paintball just barely gets over the billboard at the highest point in its trajectory.

Part A: At what angle ? above the horizontal do you need to shoot the ball to clear the billboard?

Part B: What is your horizontal distance from the billboard?

Part C: How long does the paintball take to move from the highest point in its trajectory to the height of the roof?

Part D: What is the speed of the ball when it strikes?

Part E: Does the ball strike the roof?

Explanation / Answer

For the ball to just barely get over the billboard, its vertical velocity will be 0 m/s at this height. To determine the distance the ball rises subtract the height of the gun from the height of the billboard.

d = 67.5 – 1.5 = 66 meters
Let’s use the following equation to determine the ball’s initial vertical velocity.


vf^2 = vi^2 + 2 * a * d, vf = 0, a = -9.8, d = 66
0 = vi^2 + 2 * -9.8 * 66
vi^2 = 1293.6
vi = 1293.6

The ball’s initial vertical velocity is approximately 36 m/s. Let’s use this number in the following equation to determine the angle.


Vertical velocity = v * sin
42 * sin = 1293.6
sin = 1293.6 ÷ 42 = 0.856348839
= 58.90906964 above horizontal

Let’s use the ball’s initial vertical velocity in the following equation to determine the time for the ball to reach this height.

vf = vi – a * t, vf = 0, vi = 1293.6, a = 9.8
0 = 1293.6 – 9.8 * t
t = 1293.6 ÷ 9.8

This is approximately 3.67 seconds. Let’s use this time and the ball’s initial horizontal velocity in the following equation to determine the horizontal distance from you to the billboard.

d = v * t
v = v * cos
To determine the exact value of cos , let’s use the following equation.

Sin^2 + Cos^2 = 1
Sin = 1293.6 ÷ 42
Sin^2 = 1293.6 ÷ 42^2

1293.6 ÷ 1764 + Cos^2 = 1

Cos^2 = 1 – 1293.6 ÷ 1764 = 0.266666667
Cos = 0.516397779

d = 42 * 0.516397779 * 1293.6 ÷ 9.8 = 79.59899497 meters

OR

d = 42 * cos 58.90906964 * 1293.6 ÷ 9.8 = 79.59899497 meters
Since both answers are the same, I believe that the angle and horizontal correct.

To determine if the ball lands the roof, let’s use the ball’s horizontal velocity in the following equation.

d = v * t
v = 42 * cos = 42 * 0.516397779 = 21.71306504 m/s

Let’s use the following equation to determine the time for the ball to move from the top of billboard to the top of the roof.


d = vi * t + ½ * a * t^2
d = 67.5 – 51.8 = 15.7 meters
vi is the ball’s vertical velocity at the top of the billboard. This is 0 m/s.
a = 9.8

15.7 = ½ * 9.8 * t^2
t = (15.7/4.9)
This is approximately 1.79 seconds

d = 21.71306504 * (15.7/4.9)
This is approximately 38.9 meters.
Since this number is less than 40 meters, the ball will land on the roof.

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