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.
t = 1.8
x = 36 meters
y = -3.75 meters
Vox = x/t = 36/1.8
Vox = 20 m/s
Vox = VocosƟ
20 = VocosƟ
Vo = 20/cosƟ
Voy = Vo + at
y = Voyt + 0.5at2 = -VosinƟ -4.9t2
-3.75 = (20/cosƟ)sinƟt -4.9t2 <----plug in t = 1.8
-3.75 = (20)(2)tanƟ - (4.9)(3.24)
Ɵ = 16.86 degrees
To find the direction of the balloon's initial velocity, you need to determine the angle it makes with the horizontal.
Let's break down the problem and go step by step:
Step 1: Define the given values:
- Distance between the buildings (horizontal distance) = 36.0 m
- Height of Jackson dorm = 14.5 m
- Height of Walton dorm = 21.5 m
- Time taken for balloon to hit Walton dorm = 1.8 s
Step 2: Calculate the vertical distance traveled by the balloon:
Since the balloon hits the Walton dorm halfway between the ground floor and the roof, we can calculate the vertical distance as the average of the two heights: (14.5 m + 21.5 m) / 2 = 18.0 m
Step 3: Determine the initial vertical velocity of the balloon:
Using the equation of motion for vertical motion: Δy = v0y * t + (1/2) * a * t^2
Assuming the initial vertical velocity (v0y) is upwards, Δy (vertical distance) = 18.0 m, and the time (t) = 1.8 s, we can solve for v0y.
Now, since there is no vertical acceleration (ignoring air resistance), the equation simplifies to: Δy = v0y * t
Substituting the values: 18.0 m = v0y * 1.8 s
Simplifying the equation, v0y (y-component of the velocity) = 10.0 m/s upwards.
Step 4: Determine the horizontal velocity of the balloon:
Since the horizontal distance between the buildings is known (36.0 m) and the time taken (1.8 s) is given, we can use the formula for horizontal motion: Δx = v0x * t
Substituting the values: 36.0 m = v0x * 1.8 s
Simplifying the equation, v0x (x-component of the velocity) = 20.0 m/s
Step 5: Calculate the magnitude of the initial velocity:
Using the Pythagorean theorem, the magnitude of the initial velocity (v0) can be found:
v0^2 = v0x^2 + v0y^2
Plugging in the values, v0^2 = (20.0 m/s)^2 + (10.0 m/s)^2
Simplifying, v0^2 = 400 m^2/s^2 + 100 m^2/s^2
v0^2 = 500 m^2/s^2
Taking the square root, v0 = √500 ≈ 22.4 m/s
Step 6: Determine the angle (θ):
To find the angle θ, we can use the equation: θ = tan^(-1)(v0y / v0x)
Substituting the values: θ = tan^(-1)(10.0 m/s / 20.0 m/s)
θ = tan^(-1)(0.5)
θ ≈ 26.6 degrees
Therefore, the direction of the balloon's initial velocity is approximately 26.6 degrees above the horizontal.