The soccer player kicks the ball from ground level such that its initial velocit

Question

The soccer player kicks the ball from ground level such that its initial velocity makes an angle of 20 degree with the horizontal. The soccer player is attempting to use the soccer ball to break a window 60 ft away. The window sill is located 2 ft above ground level while the window head is located 6 ft above ground level. Find the largest and smallest values of v_o, the initial speed of the soccer ball, so that it hits the window. For each value of the initial speed, find the time required for the ball to hit the window and the maximum elevation the ball achieves. Neglect the size of the ball (treat it as a particle) and air drag.

Explanation / Answer

For hitting the window sill :

In the horizontal direction : vo*cos(20)*t1 = 60

So, t1 = 60/(vo*cos(20 deg)) = 63.9/vo

In the horizontal direction : 2 = vo*sin(20)*t1 - 0.5*32.2*t1^2

So, 2 = vo*sin(20 deg)*63.9/vo - 16.1*(63.9/vo)^2

So, vo = 57.5 ft/s <------ answer

So, time required , t1 = 63.9/vo = 63.9/57.5 = 1.11 s <------answer

So, maximum elevation , h = (vo*sin(20))^2/2g

= (57.5*sin(20 deg))^2/(2*9.8)

= 19.7 ft <------ answer

For hitting the window head :

In the horizontal direction : vo*cos(20)*t2 = 60

So, t2 = 60/(vo*cos(20 deg)) = 63.9/vo

In the horizontal direction : 6 = vo*sin(20)*t2 - 0.5*32.2*t2^2

So, 2 = vo*sin(20 deg)*63.9/vo - 16.1*(63.9/vo)^2

So, vo = 64.4 ft/s <------ answer

So, time required , t2 = 63.9/vo = 63.9/64.4 = 0.99 s <------answer

So, maximum elevation , h = (vo*sin(20))^2/2g

= (64.4*sin(20 deg))^2/(2*9.8)

= 24.8 ft <-------answer

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