A Stable Physics Demo

I was wondering about the following idea for a physics demo.

You drop a metal ball from the ceiling along a vertical ramp which guides the ball and launches it at the bottom at some angle. Then you set up a small basket to catch the metal ball a certain distance away. When you do the math you get the following formula for the distance d that the basket should be away from the launch point:

d = 2 h sin 2θ

where d is the distance between the launch point and the basket, h is the original height of the ball, and θ is the launch angle from the horizontal.

In particular, if you set up the launch angle as 45 degrees, you get the nifty formula:

d = 2 h

meaning that the horizontal distance is exactly twice the height!

The last formula makes it extremely easy to set up (or to try to verify as a lab experiment). The question I had was whether this will actually work? In particular, when you set up the demo, you're bound to make small mistakes in the set up. You'll lose the "WOW" factor if your metal ball doesn't fall into the basket on your first attempt. The most brittle part of the set up is the launch angle. So it's worthwhile to compute the error introduced into d from an error in θ. I.e., how far will the ball fly off the mark if you set up the launch angle as 44 degrees instead of 45.

It turns out, that since sin 2θ has a maximum at 45 degrees, the error is actually quite small. If you do the calculus of variations you'll get

Δd / d ≈ -4 (Δθ)²

so that the proportional error in distance varies quadratically with the angular error. For an error of 1 degree and a distance of 5 m, this is less than 1 cm error -so we can expect a bullseye! For an error of 5 degrees and a distance of 5m, this is around 15 cm, so that there is a high probability of getting into a small trash can.

Even though the most brittle part of the set up is the launch angle, there are some other issues I could see interfering with perfect results:

• The angle of the launcher on the horizontal plane. You could get the right total distance but still be off the mark because you end up on left or right of target.
• Measuring h and d accurately.
• Friction on the way down or during launch that will reduce the ball's kinetic energy and hence reduce d.
• To a lesser extent, air resistance.

I'll try this experiment and report back!