In real life, collisions are often messy affairs
which require a mixture of techniques to understand.
As an example, consider the **ballistic pendulum**.

You can break the action into two pieces. Please write an equation for each piece:

- A ball of mass
**m**moving at speed**v1**slams into an arm of mass**M**. Use conservation of momentum in this completely inelastic collision to write an equation relating**v1**to the velocity of the arm-plus-ball**v2**. - An arm with an initial speed swings up until it halts
at an angle
**theta**. Use conservation of energy to figure out the relationship between the speed of the arm-plus-ball**v2**, the angle**theta**, and the height**h**by which the center of mass of arm-plus-ball rises.

Your first job today is to figure out the speed **v1** with which the
ball is fired from the catapult, by measuring the angle **theta**
(and some other quantities).

- be very gentle with the swinging arm
- use the stuffing rods to push the ball into the catapult
- make 3 measurements for the
**"medium-range" setting**of the spring-loaded gun - you will need to find the position of the center of mass of the arm-plus-ball
- watch the device when you fire the catapult. If it slides across the table, your measurements will be inaccurate: some of the momentum goes into the motion of the whole device, which you cannot measure
- try to estimate the uncertainty in each of the
**four**quantities you need to measure

- You should end up with one table showing all your measurements, and the uncertainty associated with each measurement.
- Which measurement has the largest percentage uncertainty? If you want to improve this experiment, you should concentrate on doing this part of it more precisely.

Now, show step-by-step how to use
your measurements to compute
the value of
**v1,**
the velocity of the ball when it leaves
the muzzle of the gun.

- What is this velocity for the medium-range setting?

Compute the uncertainty in **v1**,
using the rules for propagating uncertainties.

- What is the uncertainty in the velocity for the medium-range setting?

As you try to convert the uncertainties in the values you measured in class into an uncertainty in the initial speed

v1of the ball, you may find it useful to peruse

- this GENERAL guide to using uncertainties in calculations
- this SPECIFIC example of using uncertainties in the ballistic pendulum; the numbers will be different than yours, but the method should be the same

Your second job is to calculate the spring constant **k**
of the spring in the gun.
Let's use an energy-based measurement:

- Use your measurements of the initial speed of the ball to compute the kinetic energy of the ball. Assume that this kinetic energy (KE) is exactly the same as the spring potential energy (SPE) stored in the compressed spring.
- Measure the distance by which the spring is compressed for the medium-range setting. Include an uncertainty with the distance.
- Compute the spring constant using data from the medium-range setting
- Compute an uncertainty in the spring constant for the medium-range setting