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 $$53.0^{0}$$ 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.

Answer 1

(a) From the particle under constant velocity model in the $$x$$ direction, find the time at which the ball arrives at the goal:
$$x_{f}=x_{i}+v_{i}t\rightarrow t=\frac{x_{f}-x_{i}}{v_{xi}}=\frac{36.0-0}{(20m/s)\cos53.0^{0}}=2.99s$$
From the particle under constant acceleration model in the $$y$$ direction, find the height of the ball at this time
$$y_{f}=y_{i}+v_{yi}t+\frac{1}{2}a_{y}t^{2}$$
$$y_{f}=0+(20.0m/s)\sin53.0^{0}(2.99s)-\frac{1}{2}(9.80m/s^{2})(2.99s)^{2}$$
$$y_{f}=3.94m$$
Therefore, the ball clears the crossbar by
$$3.94 m − 3.05 m = 0.89 m$$
(b) Use the particle under constant acceleration model to find the time at which the ball is at its highest point in its trajectory
$$v_{yf}=v_{yi}-gt\rightarrow t=\frac{v_{yf}-v_{yf}}{g}=\frac{(20.0m/s)\sin53.0^{0}-0}{9.80m/s^{2}}=1.63s$$
Because this is earlier than the time at which the ball reaches the goal, the ball clears the goal on its way down.