The Monkey Hunter

The goal of this demonstration is to physically see a point of intersection of a system of equations. The instructor has an apparatus that releases an object (referred to as a monkey, even thought it is not a monkey) as soon as a bullet is fired at that object. If the gun is aimed correctly and if the speed of the bullet is fast enough, the bullet will hit the monkey no matter what the distance is between the gun and the monkey and the height of the monkey.

Part 1: Measure the height (which we will denote by H) of the monkey and the horizontal distance (which we will denote by L) between the apparatus holding the monkey and the gun. Using the Pythagorean theorem for right triangles, find the distance between the gun and the monkey. We will denote this distance by D.

When the bullet hits the monkey, the bullet will have traveled a horizontal distance of L. The monkey doesn't move horizontally as time elapses, but the bullet does. The horizontal distance that the bullet travels depends on the velocity (v) that the bullet was fired and the distances that you found above. The formula for the horizontal distance traveled by the bullet is given by

x = (L/D) vt

Since we are interested in when this horizontal distance is L, we want to find the time when x=L.

Part 2: Substitute the values for L and D that you found in part 1 above into the two

equations

x = (L/D) vt

and

x = L

Then set these two equations equal to each other and solve for t.

Another thing that we can notice about this system is that when the bullet hits the monkey the height of the bullet and the monkey must be equal. Recall from an earlier demonstration that the height of an object after t seconds that is dropped from a given height H is given by

y = H - (1/2)gt²

where g is the constant due to gravity. The height of the bullet after t seconds is given by

y = (H/D)vt - (1/2)gt²

Notice that not only is gravity affecting the height, but the initial velocity of the bullet also affects the height.

Part 3: Substitute the values for H, g, and D that you found in part 1 above into the two equations

y = H - (1/2)gt²

and

y = (H/D)vt - (1/2)gt²

Then set these two equations equal to each other and solve for t.

Notice that in both cases we got the equation t=D/v. So, if we know the distance D and the velocity v at which the bullet is fired, we can find the time it takes for the bullet to hit the monkey. Once we have a value for t we can substitute t back into the equations for x and y to find the horizontal and vertical height of the object when it is struck by the bullet.

Part 4: Substitute the value t = D/v into the two equations that we had describing the horizontal distance for x. What is the value for x in each equation? Why should the x values be the same regardless of which equation you picked?

Part 5: Substitute the value t = D/v into the two equations that we had describing the vertical distance for y. What is the value for y in each equation? Why should the y values be the same regardless of which equation you picked?