Solution

• Identify the relevant pieces of information in the problem. The initial positions and velocities of the monkey and Z are relevant. We're interested in whether the projectile and monkey collide. That is, is there a time when the coordinates of the monkey equal the coordinates of the projectile? So the coordinates as a function of time are also relevant.
• Give names to all the relevant variables. Place Z at the origin. It's always a good idea to make the simplest possible choice of coordinate system. Call the initial position of the monkey . Call The initial velocity of the projectile . The initial velocity of the monkey is zero, since it's dropping from rest. Call the position vectors of the monkey and Z at a time t, and respectively.
• Identify what equations describe the physical situation we are analyzing. Both the projectile and the monkey are in free fall so we can use eq. 5.13 to describe both of them.

The strategy we'll take to solve this problem is to write down the two equations that describe the coordinates of the monkey and projectile and then set then equal. We set the coordinates equal because that means the the projectile and the monkey have collided. This will tell us what condition must satisfy in order to hit the monkey.

OK, so let's write down the two equations:

and

Now let's set these equations equal. Notice that the terms cancel. So the acceleration (i.e. gravity) cancels out completely. So at this point there are two ways to proceed

1. Well if gravity doesn't matter, just take it to be zero from the beginning. The monkey could be in deep space where there is no gravity. If it tries to drop now, it will discover that it's not dropping. So if you aim directly at the monkey, you'll be bound to hit it!
2. Let's look at the equation we get: . That says that the vectors and have the same direction. So if Z wants to hit the monkey, the gun should be pointed directly at it.

Either way you see that Z will hit the monkey by aiming directly at it, despite the fact that it is falling. The point is that the decrease in height due to gravity is exactly the same for both objects, , despite the fact the bullet is going much faster than the monkey. So gravity is irrelevant to the relative position of two objects in free fall.