Problem: A golfer (a fat penguin) on NGC is on the fairway of the fourth hole. Unfortunately, there is a hill of 20 m height between him and the green. He is 33 m from the centerline of the hill and the center of the green is 22 m from the hill's centerline. The green is roughly circular, with a diameter of 3.5 m. What initial velocities and angles are necessary for him to hit the green? Hint: Assume that his initial speed is fixed at 40 m/s and he can vary only the initial firing angle (by choosing various clubs).

Solution:  Taking the initial position of the ball as the origin, x_o = y_o = 0. I will take up and to the right as positive, so that the acceleration in the y-direction is -9.8 m/s². Since the ball lands at the same level that it starts at, the displacement in the y-direction is zero. To hit the green, the range x has to be between 53.25 and 56.75 m. At an x-displacement of 33 m, the y-displacement has to be a minimum of 20 m.. Writing the equations of constant acceleration in both directions,

Substituting 40 m/s for the initial speed, and using the distance to the near end of the green,

Doing this again for the distance to the far edge of the green,

It is possible that all four angles will clear the hill. We have to check. Maybe none will work.

At 33 m, the height must be a minimum of 20 m:

The next step is to try each of the above angles. For 9.52^o and 10.2^o, the heights that would be reached at a range of 33 m is 2.11 and 2.49 m, respectively. So these angles will not clear the hill, as you probably suspected. For 80.5^o and 79.8^o, the heights reached are 74.8 and 77.1 m, respectively.

Therefore, if the fat penguin uses an initial angle of 80.5, he'll hit the near edge of the green; if he uses 79.8, he'll hit the far edge of the green.

By the way...Why do penguins have green eyes? To hide on greens, of course! You've never seen a penguin on a green? They hide good, don't they?