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

∅=10.7°

Step-by-step 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=(x)/(v_(0)cos\theta)=(45)/(30cos\theta)`

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

`y=v_(0)sin\theta.t-(1)/(2)a.t^(2) =30sin\theta.t-(1)/(2)9.8t^(2)`

replacing time.

`2.90=30sin\theta((45)/(30cos\theta))-4.9((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.