A place-kicker must kick a football from a point 36.0 m (about 40 yards) from th

Question

A place-kicker must kick a football from a point 36.0 m (about 40 yards) from the goal. Half the crowd hopes the ball will clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 23.5 m/s at an angle of 52.5° to the horizontal. (a) By how much does the ball clear or fall short of clearing the crossbar? (Enter a negative answer if it falls short.) (b) Does the ball approach the crossbar while still rising or while falling?

(a) The football is moving with constant velocity in the x-direction. We denote the initial horizontal position of the football by x0 and its initial x-component of velocity by v0x. For the position in the horizontal direction in terms of time t, we have

x = x0 + v0xt.

Solving for t gives the following equation.

t = X-Xo / VoX

We do not know the horizontal component of velocity but we can calculate it from the angle of the football's path to the horizontal. Remember that we use trigonometry to find horizontal and vertical components of a vector when we know its magnitude at a particular angle. We have

VoX=( ? m/s ) cos ( ? )*= ( ? )m/s.

Explanation / Answer

(a)

initial vertical velocity = 23.5*sin(52.5) = 18.64

initial horizontal velocity =23.5*cos(52.5) = 14.30

time to reach the crossbar(t) = 36/23.5cos(52.5) = 2.51 s

height reached at this time = 23.5sin(52.5)*2.51 -.5*g*(2.51)^2 = 46.98 - 30.87 = 16.11

it clears by a distance of 16.11 -3.05 = 13.06 m

(b)

time to reach max height = 18.64/g = 18.64/9.81 = 1.90 s

ball approaces crossbar while falling (after max height as t > 1.90s)

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