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 22.6 m/s at an angle of 47.0° 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 falling

Answer 1

Answer:

(a). H = 8.83 m

(b). Rising

Explanation:

Given,

• Initial velocity of the ball = u = 22.6 m/s
• Height of the crossbar = h = 3.05 m
• Distance of the crossbar from the initial position of the ball = r = 36.0 m
• Angle of projection = $\theta\ =\ 47.0^o$
• Initial horizontal velocity of the ball =$u_x\ =\ ucos\theta$
• Initial vertical velocity of the ball = $u_y\ =\ usin\theta$

part (a)

In the horizontal direction,

Ball is moving with the constant initial velocity of $u_x$

Let 't' be the time of the ball to reach at the crossbar of distance r.

$\therefore r = u_xt\\\Rightarrow t\ =\ \dfrac{r}{u_x}\ =\ \dfrac{r}{ucos\theta}\\\Rightarrow t\ =\ \dfrac{36.0}{22.6\times cos47^o}\\\Rightarrow t\ =\ 2.33\ sec$

At time 't', let 'y' be the vertical displacement attained by the ball.

From the equation of kinematics,

$y\ =\ u_yt\ -\ \dfrac{1}{2}gt^2\\\Rightarrow y\ =\ usin\theta t\ -\ \dfrac{1}{2}gt^2\\\Rightarrow y\ =\ 22.6sin47^0\times 2.33\ -\ 0.5\times 9.81\times 2.33^2\\\Rightarrow y\ =\ 11.88\ m$

Total distance of the ball above the crossbar = H = y - h = 11.88 - 3.05 = 8.83 m

part (b)

At the maximum height, vertical velocity of the ball becomes zero,

Let h be the maximum height attained by the ball.

From the kinematics,

$\therefore (v_y)^2\ =\ (u_y)^2\ -\ 2gh\\\Rightarrow 2gh\ =\ (u_y)^2\\\Rightarrow h\ =\ \dfrac{(u_y)^2}{2g}\\\Rightarrow h\ =\ \dfrac{(22.6\times sin47^o)^2}{2\times 9.81}\\\Rightarrow h\ =\ 13.92\ m$

Maximum height attained by the ball with the given initial velocity is 13.92 m but at the crossbar it attains 11.88 m. Hence the ball will still rising approaching the crossbar.