Answer to Question #143167 in Physics for Erica

Question

In the last five seconds of the championship match between teams A and B the scores were tied up at 75 when player from team A was fouled and was awarded two free throws . On the first attempt, he released the ball with velocity of 6.0 m/s at 45 degree above horizontal from a height of m from the floor (a) is this a miss or basket? (b) for the second throw, the player shoots against at 45 degree from the same point of release. What must be the initial speed of the ball so that it will go through the basket ? The center of the basket is 4.21 m from the free throw line and 3.05m above the floor .

Answer 1

Draw what's going on:

(a) The ball rises up during a time of

$$t_1=\frac{v_y}{g}=\frac{v\text{ sin}\theta}{g}.$$

The maximum height above the blue ground level is

$$H=\frac{v_y^2}{2g}=\frac{v^2\text{ sin}^2\theta}{2g}=0.918\text{ m}.$$

At 6 m/s thrown at 45° above the horizontal, the ball will rise up for 91.8 cm.

Then the ball must fall into the basket a distance of

$$d=H-(L-h),$$

which will take a time

$$t_2=\sqrt{\frac{2d}{g}}=\sqrt{\frac{2(H-(3.05-h))}{g}}.$$

In this equation, h is the height above the orange ground at which the ball was thrown.

Meanwhile, the horizontal range must be 4.21 m:

$$4.21=R=v_xt=v_x(t_1+t_2)=v\text{ cos}\theta(t_1+t_2),\\\space\\ R=v\text{ cos}\theta\bigg[\frac{v\text{ sin}\theta}{g}+\sqrt{\frac{2(H-(L-h))}{g}}\bigg].$$

Check whether this will happen if the ball was thrown from 2.2 m above the orange ground:

$$R=6\text{ cos}45°\bigg[\frac{6\text{ sin}45°}{9.8}+\sqrt{\frac{2(0.918-(L-2.2))}{9.8}}\bigg]=\\\space\\=1.83\text{ m}.$$

This is a miss.

(b) We have our range of 4.21 m, the velocity must be of such a value that will make the range 4.21 m. Plug H into the equation for R:

$$R=v\text{ cos}\theta\Bigg[\frac{v\text{ sin}\theta}{g}+\sqrt{\frac{2\big(\frac{v^2\text{ sin}^2\theta}{2g}-(L-h)\big)}{g}}\Bigg],\\\space\\ R=\frac{v^2\text{sin}(2\theta)}{2g}+\frac{\sqrt{v^2\text{ sin}^2\theta-4g(L-h)}}{g}.$$

Substitute the height of the throw and we have only one undefined, v. Solve this equation for v with any method you like. For example, if h=2.2 m, the value of v is 8.82 m/s.