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 23.9 m/s at an angle of 51.5 degrees 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?

Answer 1

Explanation:

Let's calculate the components of the football's velocity:

$v_{0x} = (23.9\:\text{m/s})\cos{51.5°} = 14.9\:\text{m/s}$

$v_{0y} = (23.9\:\text{m/s})\sin{51.5°} = 18.7\:\text{m/s}$

a) The time it takes for the football to travel 36.0 m horizontally is

$t = \dfrac{x}{v_{0x}} = \dfrac{36.0\:\text{m}}{14.9\:\text{m/s}} = 2.4\:\text{s}$

During this time, the y-displacement of the football is

$y = v_{0x}t - \frac{1}{2}gt^2$

$\:\:\:\:= (18.7\:\text{m/s})(2.4\:\text{s}) - \frac{1}{2}(9.8\:\text{m/s}^2)(2.4\:\text{s})^2$

$\:\:\:\:= 16.7\:\text{m}$

This means that the football cleared the crossbar by 16.7 m - 3.05 m = 13.7 m

b) To determine whether the football was rising or falling while clearing the crossbar, let's look at the y-component of its velocity after 2.4 s:

$v_y = v_{0y} - gt = 18.7\:\text{m/s} - (9.8\:\text{m/s}^2)(2.4\:\text{s})$

$\:\:\:\:\:\:= -4.82\:\text{s}$

Since its sign is negative, this means that the football was already on its way down.