# Math Genius: 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.

If \$g\$ denotes the gravity, \$u\$ the initial speed, \$d\$ the travelled horizontal distance, \$xin left[0,fracpi2right]\$ the angle to the ground of the initial velocity, and \$t\$ the flight time, then you have \$u cos(x) t= d\$ and \$usin(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}{ucos(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.

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