Question

A football kicker can give the ball an initial speed of 30 m/s. He is 45 m from the goalpost which has a crossbar 2.90 m high. What is the smallest angle of elevation that he can kick the football and score a field goal?

Answer 1

Answer:

∅=10.7°

Explanation:

In parabolic motion the position on the x-axis can be found like this.

$x=V_{0}.cos \theta.t$

Where we clear the time.

$t=\frac{x}{v_{0}cos\theta}=\frac{45}{30cos\theta}$

The position on the y-axis can be found as well.

$y=v_{0}sin\theta.t-\frac{1}{2}a.t^{2} =30sin\theta.t-\frac{1}{2}9.8t^{2}$

replacing time.

$2.90=30sin\theta(\frac{45}{30cos\theta})-4.9(\frac{45}{30cos\theta})^{2}$

$2.9=45tan\theta-11.025sec^{2}\theta=45tan\theta-11.025(1+tan^{2}\theta)$

$-11.025tan^{2}\theta+45tan\theta-8.125=0$

Now we use the quadratic equation to find the tangent of the angle.

$tan\theta=3.89\\ and\\ tan\theta=0.19$

Finally we use the arc tangent function to find the angle.

$\theta=75.6\\ and\\ \theta=10.7$

We choose the second angle because it adapts to the situation described, that is the minimum angle.