Predict the landing spot of a ball

Your job in this exercise is to watch very carefully as a ball is shot out of a gun, flies through the air, and lands on the floor. After you have analyzed its motion, you must predict where the ball will land when shot from a slightly different location.

Here's the basic setup:

I'll place one of the ballistic pendulum devices on the table at the front of the room. We'll write the values of the heights h1, h2 on the board, then fire gun. I'll measure the distances x1, x2 and write those on the board, too.

  1. Determine the uncertainty in the total height and total horizontal distance.

  2. Calculate the initial speed v of the ball when it leaves the muzzle of the device. Explain to me in writing your method, and show all your work.
  3. Determine the uncertainty in v.

  4. Write an equation which describes the range of the ball when fired from any height H. So, for example, if I placed your gun 2 meters above the floor, or 5 meters, or 10 meters, how far would the ball go?

  5. Predict the distance the ball will travel if shot from one of the student tables. I'll measure the height of the table -- ask me for the value and its uncertainty. You must show me or the assistant your result before you can continue! If you want full credit, you'll again have to deal with uncertainties, and end up by showing me a range: "the ball will travel 156 +/- 3 cm", or something like that.

Near the end of the class, I'll move the gun to one of the student tables and fire it. Your score will depend on how well your prediction and its uncertainty matches the actual distance.

How do you deal with combining two measurements, each of which has an associated uncertainty?