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 20.0 m/s at an angle of 53.0° to the horizontal.

(a) By how much does the ball clear or fall short of clearing the crossbar?

(b) Does the ball approach the crossbar while still rising or while falling?

Short answer

1. 0.9m
2. The ball clears the goal while falling

Step-by-step solution

Taking the horizontal and vertical components of initial velocity

$u_x=u\cos \theta =20\cos 53°=12.036m/s$

$u_y=u\sin \theta =20\sin 53°=15.97m/s$

Solving for time “t”

Taking the horizontal components

$x=x_i+u_{x}t+\frac{1}{2}{a}_x{t}^2$

$36=0+12.036t+0$

$t=2.99s$

Solving for vertical distance covered in time t

Taking vertical components

$y=y_i+u_{y}t+\frac{1}{2}{a}_y{t}^2$

$y=0+15.97\times 2.99+\frac{1}{2}(-9.8)(2.99{)}^2$

$y=47.75-43.8$

$y=3.95m$

Solving for the difference in height (a)

Since the crossbar’s height $=3.05m$

The ball clears the crossbar by $3.95-3.05=0.9m$

Checking if the ball clears the goal while falling or rising

Since,

$v_y=u_y+a_{y}t$

$0=15.97-9.8t$

$t=1.629s$

Since the ball reaches the goal at$t=2.99s$ and it reaches its highest point at$t=1.629s$ .

Hence, the ball clears the goal while falling.