In the annual battle of the dorms, students gather on the roofs of Jackson and Walton dorms to launch water balloons at each other with slingshots. The horizontal distance between the buildings is 36.0 m, and the heights of the Jackson and Walton buildings are, respectively, 14.5 m and 21.5 m. Ignore air resistance.
(a) The first balloon launched by the Jackson team hits Walton dorm 1.8 s after launch, striking it halfway between the ground floor and the roof. Find the direction of the balloon's initial velocity. Give your answer as an angle measured above the horizontal.
To find the direction of the balloon's initial velocity, we can use the equations of projectile motion.
First, let's separate the motion into horizontal and vertical components.
Horizontal motion:
The horizontal distance between the buildings is 36.0 m, and the time it takes for the balloon to reach the target is 1.8 s. We can use the equation:
distance = velocity × time
36.0 m = (Vx)(1.8 s)
Solving for Vx, we get:
Vx = 36.0 m / 1.8 s
Vx = 20.0 m/s
Vertical motion:
The balloon hits Walton dorm halfway between the ground floor and the roof. The height of Walton building is 21.5 m, so the balloon reaches a height of (21.5 m / 2) = 10.75 m above the ground.
To find the initial vertical velocity (Vy), we can use the equation:
final height = initial height + (initial velocity) × (time) + (1/2) × (acceleration) × (time^2)
Plugging in the values, we get:
10.75 m = 0 + (Vy)(1.8 s) + (1/2)(-9.8 m/s^2)(1.8 s)^2
Simplifying the equation, we have:
10.75 m = 1.8 s(Vy) - 15.876 m
Rearranging the equation, we get:
1.8 s(Vy) = 10.75 m + 15.876 m
1.8 s(Vy) = 26.626 m
Dividing both sides by 1.8 s, we get:
Vy = 26.626 m / 1.8 s
Vy = 14.792 m/s
Now that we have the horizontal and vertical components of the initial velocity (Vx and Vy, respectively), we can use the trigonometric relationship between the sides of a right triangle to find the angle θ.
The tangent of an angle θ is equal to the ratio of the opposite side (Vy) to the adjacent side (Vx):
tan(θ) = Vy / Vx
tan(θ) = 14.792 m/s / 20.0 m/s
Using a calculator, we can find the inverse tangent of this ratio:
θ = arctan(14.792 m/s / 20.0 m/s)
θ ≈ 37.7°
Therefore, the direction of the balloon's initial velocity is approximately 37.7° above the horizontal.