# Find the initial direction and time of flight of a basketball, given initial speed and distance

A player passes a basketball to another player who catches it at the same level from which it was thrown. The initial speed of the ball is 7.1m/s, and it travels a distance of 4.6m. What were (a) the initial direction of the ball and (b) the time of flight?

I can't figure a way using only kinematic equations and soh cah toa, am I missing something? I tried using trigonometric identities but got stumped late into the algebra.

• Is 4.6 meters the distance travelled in air, or the horizontal displacement? The latter interpretation makes the problem much simpler. – Semiclassical Oct 18 '15 at 0:13
• I didn't understand that, what's the latter interperatation – Zayn Malek Oct 18 '15 at 0:17
• It's what @C.I.J. assumes in his solution – Semiclassical Oct 18 '15 at 0:20

If $g$ denotes the gravity, $u$ the initial speed, $d$ the travelled horizontal distance, $x\in \left[0,\frac\pi2\right]$ the angle to the ground of the initial velocity, and $t$ the flight time, then you have $u \cos(x) t= d$ and $u\sin(x) t-\frac12gt^2=0$. Solve for $x$ to get $\sin(2x)=\frac{gd}{u^2}$. There are two possible values for $x$, whence also two values for $t=\frac{d}{u\cos(x)}$.
Is the ball is thrown along a parabolic path at an angle $\alpha$ with speed $v$, the time is
$$\frac{distance }{v \cos \alpha}$$
So $\alpha$ should also be known.