Question

A placekicker 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 23.4 m/s at an angle of 48.0° 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.)
m
(b) Does the ball approach the crossbar while still rising or while falling?

rising or falling

Answer 1

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Let B = angle above horizontal for the kick.

v0 = 23.4 m/s

v sub y0 = ( v0 ) ( sin B ) = ( 23.4 ) ( sin 48.0 ) = 17.389 m/s

v sub x0 = ( v0 ) ( cos B ) = ( 23.4 ) ( cos 48.0 ) = 15.657 m/s

Consider travel in y direction ( vertical direction ) :
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a sub y = d v sub y / dt = -g

d ( v sub y ) = ( -g ) dt

v sub y = v sub y0 - gt

dy/dt = v sub y = v sub y0 - gt

dy = [ v sub y0 - gt ] dt

y = ( v sub y0 ) ( t ) - ( g/2 ) ( t^2 )

Let t* be time for: y = y* = ymax and v sub y = 0.0 :
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t* = v sub y0 / g

t* = v sub y0 / g

t* = 17.389 / 9.807 = 1.773 s

y* = ( v sub y0 ) ( t* ) - ( g/2 ) ( t* )^2

y* = ( 17.389 ) ( 1.773 ) - ( 9.807 / 2 ) ( 1.773 )^2

y* = 30.830697 m - 15.4142 m = 15.4164 m

total flight time = t** = ( 2 ) ( t* ) = ( 2 ) ( 1.773 ) = 3.546 s

Consider the horizontal direction :
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v sub x = v sub x0 = 15.657 m / s

xf = ( v sub x0 ) ( tf )

tf = xf / v sub x0 = ( 36.0 m ) / ( 15.657 m /s ) = 2.299 s

We now need yf corresponding to t = tf :
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yf = ( v sub y0 ) ( tf ) - ( g / 2 ) ( tf )^2

yf = ( 17.389 ) ( 2.299 ) - ( 9.807 / 2 ) ( 2.299 )^2

yf = 39.977 - 25.91696 = 14.0603 m

yf - ycb = 14.0603 - 3.05 = 11.0103 m <-------------

The field goal try is successful.