Aplayer kicks a football (from the ground) at an initial angle of 30°. the football is in the air for 2.4 s before it hits an opposing player. the opposing player was a horizontal distance x = 50 m away from where the football was kicked. what was the initial horizontal velocity component of the football?

The initial horizontal velocity was 21 m/s

Explanation:

Please, see the figure for a better understanding of the problem.

The equation for the position of an object moving in a parabolic trajectory 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

Notice that at time t = 2.4 s the vector "r" is the one in the figure. We know that the x-component of that vector is 50 m. Then using the equation for the x-component of the vector "r", we can calculate the initial velocity:

x = x0 + v0 · t · cos α

Let´s place the center of the frame of reference at the point of the kick so that x0 = 0.

x =  v0 · t · cos α

x/t = v0 · cos α

Notice in the figure that v0 · cos 30° = v0x which is the initial horizontal velocity. Remember trigonometry of right triangles:

cos α = adjacent / hypotenuse = v0x / v0

Then:

50 m/ 2.4 s = v0 · cos 30° = v0x

v0x = 21 m/s

the independent variable, the one that is intentionally manipulated, is whether or not the car hits the barrier.

the dependent variable, the one that you measure the response in, is the distance that the washers travel.