**Problem : **
A basketball player who is standing 15 feet away from a basketball hoop is
trying to make a basket. If the height of the hoop is 10 feet, and the
height at which the player shoots the ball is 6 feet, at what angle and with
what speed should the player shoot the ball?

Figure %: Diagram of a basketball player shooting a ball with speed *v* and
angle *θ*.

Using the general formula for projectile motion worked out in this section,
we find that the position function for a ball which is shot from a height of
6ft, with initial velocity vector **v**_{0}, is:

For the player to make his basket, the trajectory of the ball must pass
through the point (15, 0, 10) (since the basket is located 15 ft away and is
10 ft high). Therefore, we need to find **v**_{0} = (*v*_{x}, 0, *v*_{z}) such that
**x**(*t*) = (15, 0, 10) at some time *t* (since there is no motion in the
*y*-direction we know that *v*_{y} must be zero). To do this, we write out the
one-dimensional equations for the *x*- and *z*-directions:

(1) 15 =v_{x}t, and (2) 10 = -gt^{2}+v_{z}t+ 6

From (2) we find that the ball reaches a height of 10 ft at a time *t* such
that:

It is not enough for the ball to have a height of 10 ft, however, in order for
the basketball player to make the basket. At the same time it reaches that
height (i.e. at time *t*_{1}) it must also have traveled 15 ft in the *x*-
direction. We are now in a position to solve for *v*_{x} in terms of *v*_{z}.
Using (1), and plugging in the time *t*_{1}, we find that:

Figure %: Several solutions to the problem of a basketball player shooting
to make a basket.

**Problem : **
Assume that the basketball player in the previous problem has a special
condition which allows him only to shoot basketballs at a speed of 10 ft/s.
How might someone go about finding at what angle must he shoot the ball in
order to make the basket?

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