Consider the kinematic equation:

$\overline{)\mathbf{}{\mathit{y}}{\mathbf{=}}{\mathbf{}}{{\mathit{v}}}_{{\mathbf{0}}{\mathit{y}}}{{\mathit{t}}}_{{\mathbf{c}}}{\mathbf{-}}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{g}}{{{\mathit{t}}}_{{\mathbf{c}}}}^{{\mathbf{2}}}}$

**Part A**

Since the ball is thrown and caught at the same height, the net vertical distance covered by the ball is zero.

Now we have:

0 = v_{0y}t_{c} - (1/2)gt_{c}^{2}

v_{0y}t_{c} = (1/2)gt_{c}^{2}

A quarterback is set up to throw the football to a receiver who is running with a constant velocity *v*? r directly away from the quarterback and is now a distance *D* away from the quarterback. The quarterback figures that the ball must be thrown at an angle *?* to the horizontal and he estimates that the receiver must catch the ball a time interval *t*c after it is thrown to avoid having opposition players prevent the receiver from making the catch. In the following you may assume that the ball is thrown and caught at the same height above the level playing field. Assume that the *y* coordinate of the ball at the instant it is thrown or caught is *y*=0 and that the horizontal position of the quarterback is *x*=0. (Figure 1)

Use *g* for the magnitude of the acceleration due to gravity, and use the pictured inertial coordinate system when solving the problem.

Part A:

Find *v*0y, the vertical component of the velocity of the ball when the quarterback releases it.

Express *v*0y in terms of *t*c and *g*.

Part B:

Find *v*0x, the initial horizontal component of velocity of the ball.

Express your answer for *v*0x in terms of *D*, *t*c, and *v*r.

Part C

Find the speed *v*0 with which the quarterback must throw the ball.

Answer in terms of *D*, *t*c, *v*r, and *g*.

Part D:

Assuming that the quarterback throws the ball with speed *v*0, find the angle *?* above the horizontal at which he should throw it.

Your solution should contain an inverse trig function (entered as asin, acos, or atan). Give your answer in terms of already known quantities, *v*0x, *v*0y, and *v*0.

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