Answer:
The launch speed has to be 15.2 m/s so that the T-shirt reaches you.
Explanation:
The equation for the position of an object moving in a parabolic motion is as follows:
r = (x0 + v0 · t · cos α, y0 + v0 · t · sin α + 1/2 · g · t²)
Where:
r = vector position at time t
x0 = initial horizontal position
v0 = initial velocity
t = time
α = launching angle
y0 = initial vertical position
g = acceleration due to gravity
Please, see the figure for a description of the problem. The vector "r" is the position vector the t-shirt must have to reach you.
As you can see from the figure, the x-component of the vector "r" is 20.0 m and the y-component is 3.00 m. Then, using the equations for the vector "r" we can obtain the time of flight and the initial velocity:
x = x0 + v0 · t · cos α (since the origin of the frame of reference is located at the launching point, x0 = 0)
x = v0 · t · cos α
Solving for v0:
x/(t · cos α) = v0
20.0 m/ t · cos 45° = v0
Replacing v0 in the equation for the y-component of the vector "r":
y = y0 + v0 · t · sin α + 1/2 · g · t² (y0 = 0, same as x0)
y = 20.0 m · sin 45°/cos 45° - 1/2 · 9.80 m/s² · t²
3.00 m = 20. 0 m · tan 45° - 1/2 · 9.80 m/s² · t²
3.00 m - 20. 0 m · tan 45° = -4.90 m/s² · t²
-17.0 m/ -4.90 m/s² = t²
t = 1.86 s
Now, with the time, we can calculate v0:
v0 = 20.0 m/ t · cos 45° = 20.0 m/ 1.86 s · cos 45° = 15.2 m/s
The launch speed has to be 15.2 m/s.
