Lab 14: Impulse-Momentum Theorem

In a collision, kinetic energy is often transferred into heat and sound energy, such that it is hard to measure and not conserved. In such cases, we use conservation of momentum to model the reaction of the collision. Momentum is defined as:

p = mass * velocity

We also know that momentum is caused by force over the time interval which the force acts. We call this impulse, which is defined as:

J = Force * time

Therefore, since the momentum of the system cannot change without forces, we know that, if friction and gravity could be disregarded, since the magnitude of force before and after a collision must be 0, there is no change in momentum before and after the collision. And unlike kinetic energy which could be transferred into other energy forms, momentum, given enough time, only transfers momentum to other objects. Thus, we say that momentum of a system is conserved, regardless of elastic or inelastic collisions (which just vary in force-time). This is shown by the Impulse-Momentum equation (the change of impulse is equal to the change of momentum):

F * Δt = m * Δv

...wherein the change in momentum mΔv could also be defined as:

m * (v₂ - v₁)

If we know the force and velocity of the moving object, then we could test whether this is true. But how is this possible?!

Once again, we produce our trusty gadgets: the motion sensor and the force sensor! As pictured above, we have two wheeled carts on a level track, making the effects of friction and gravity (hopefully) negligible. We put pieces of paper under the track until it is precisely leveled, showing less than 0.4° on the phone app, such that the cart does not roll on the track by itself. Then we mount a force sensor atop it with a rubber attachment to simulate an elastic collision. The rubber would bounce off the translucent plastic on the second cart, causing force over the collision time. We measure the cart with the force sensor to be m = 433 g. The motion sensor, attached to the other end of the track, measures the velocity and position of the track.

So... we hook the sensors up to Logger Pro, and give the red cart a push, being careful the cables don't contribute tension or get in the way of the sensors. Logger Pro gives us the standard force vs time, and velocity vs time graphs:

What we're interested in here is the change of impulse, which can be obtained by using the integral function to find the curve under the force curve during the collision. On the other hand, we find the change of momentum by finding the highest and lowest velocities right before and after the collision, and multiplying that by the mass. In our first experiment, we find that our impulse is 0.5358 Ns. Our Δv, as reported by Logger Pro analysis, is 1.282 m/s. Multiplied by our cart's mass of 0.433 kg, we get the figure calculated in purple in our data table. Although it reads impulse, it is actually momentum (but also impulse!), coming in at 0.555 kg m/s. This is under 4% error, a relative success by our standards--it shows a clear correlation.

The instructions require us to run a second experiment with additional mass, and a third inelastic collision without the mass, but having messed up the inelastic collision once, we ended up doing it with the mass on. It should be inconsequential. The higher mass supposedly demonstrates that larger momentum change is conserved, while the inelastic collision increases collision time. But by the time we got to the second experiment, we've realized that the force sensor has less chance of peaking if we pushed the cart slower, therefore our second and third experiments actually showed less velocity, and therefore less momentum change.

Without further ado, we've tied a 200 g mass to the back of the cart for our next two experiments, bringing the total mass of our cart up to 633 g. Our second push yielded:

This one was pushed much slower to ensure that the force sensor captures without problem, but also to ensure that the mass stays in place. As such, the Δv is only 0.479 m/s; multiplied by 0.633 kg, the resultant change in momentum is approximately 0.303 kg m/s. Using integral again, our impulse is 0.2994 Ns. This time there's less than 2% error, not bad!

The third trial, once again, is with the additional mass, and also with a nail attachment to the force sensor, and putty on the other end, in order to create an inelastic collision. The capture looks like this:

The cart was pushed faster, but ended up stuck to the putty at 0 velocity, so the change in velocity turned out about the same as the second trial, at Δv = 0.510 m/s. Once again, using 0.633 kg mass, we calculate 0.323 kg m/s momentum change. The impulse turned out to be 0.3210 Ns. Under 1% error.

The point here is that, regardless of the mass or material of the cart, momentum and impulse are conserved, thus proving that the Impulse-Momentum Theorem is true. One unexpected result from this experiment is that the collision time are all under 1/5 of a second, and neither mass, nor the putty, seemed to greatly affect it.