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In C. S. Forster's novel Lieutenant Hornblower (set in the early 1800's) a British naval vessel tries to sneak by a Spanish garrison. The ship passes as far away from the Spanish guns as it can -- a distance s. The Spanish gunner knows that his gun has a muzzle velocity equal to v₀.

1. Once the gun is fired, what controls the motion of the cannonball? Write the equations that determine the vector acceleration of the cannonball after it leaves the cannon. You may ignore air resistance.
2. Suppose the gunner inclines his gun upward an angle q to the horizontal. Solve the equations you have written in part (a) to obtain expressions which can be evaluated to give the position of the cannonball at any time, t.
3. If the gunner wants the cannonball to hit the ship, he must choose his angle correctly. Explain how he can calculate the correct angle. (Again, you may ignore air resistance.)
4. If the muzzle velocity of the cannon is 100 m/s and the ship is a distance of 1/2 a kilometer away, find the angle the gunner should use. (Take g to be 10 m/s/s.)

You may need one or more of the following trigonometric identities (i.e., these are true for all angles, q):

    

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