27-Apr-2015: Conservation of momentum and energy

PURPOSE

The purpose of this experiment was to combine the conservation laws of momentum and energy to find an unknown velocity of a system.

PROCEDURES

Figure 1

In this experiment, we used the set-up shown in Figure 1. We pulled on the trigger (circled in green) to shoot a ball into a concave in a block (circled in red) to give the block a velocity and raise it above its initial position. The block swung on a set of strings, which made an angle with the vertical as the block was raised. This angle was measured on the part of the apparatus circled in blue. In addition, we measured the mass of the ball and the block, and the length of the strings. All the measurements are illustrated below in Figure 2. M was the mass of the block, while m was the mass of the ball. L was the length of the strings and θ was the angle that the strings made with the vertical.

Figure 2: Measurements with uncertainties

Figure 3: Derivation of initial velocity of ball

The uncertainty of each measurement was also included in Figure 2. These uncertainties were later used to find the propagated uncertainty of the ball's initial velocity. In order to find the propagated uncertainty, we had to first find an equation for the velocity. We applied the conservation of momentum equation to the system before and after the ball was embedded in the block. Initially, only the ball had momentum because the block was at rest. After the collision, the ball and the block moved as a single unit and had the same final velocity. This was supposed to mimic an inelastic or a plastic collision. Then, we applied the conservation of energy equation to the system. We took the system's initial position right after the collision as the reference point at which point it did not have any gravitational potential energy. We took the maximum height that the system reached as its final position. At this point, the system was momentarily at rest before swinging back down. Therefore, it only had a final gravitational potential energy and no kinetic energy. We then plugged in the final velocity that we found in the momentum equation as the initial velocity of the energy equation to find the resulting equation shown in Figure 3. This was the ball's initial velocity after being shot out of the spring gun, which we found to be 329.88 cm/s.

Figure 4: Partial derivatives with respect to each of the four variables

After finding an equation for the ball's initial velocity, we proceeded to take the partial derivative of the equation with respect to each of the four variables: M, m, L and θ. This process is shown in Figure 4.

Figure 5: Partial derivative with respect to M and m after plugging in values

Figure 6: Partial derivatives with respect to L and θ after plugging in values

Then, we plugged in the measured values to each of the four resulting equations (Figures 5 and 6). One thing to note is that we converted the mass measurements into kilograms and the length measurements into meters to be consistent in our calculations. Alternatively, we could have just kept the original units and converted g (the acceleration due to gravity) into cm/s² (we could have kept the masses in grams either way because they ended up canceling out). 

Figure 7: Propagated uncertainty

Finally, we multiplied the absolute values of the resulting values to the uncertainties of the corresponding measurements, as shown in Figure 7, to find the propagated uncertainty of the ball's velocity. The resulting value was in meters, which we converted back to centimeters since the ball's velocity was in centimeters as well. As it can be observed from the image, we found the propagated uncertainty to be 5.07 cm/s. We can conclude from this result that the error was not too large since the uncertainty is much smaller than the calculated velocity, which was 329.88 cm/s. 

CONCLUSION

In this experiment, we implemented our knowledge on the conservation of momentum and energy in order to find the initial velocity of an object. It was good practice in applying theoretical knowledge in a real-life situation. We exercised our ability to find the propagated uncertainty of a derived value. Practicing the steps in order to accomplish this allows us to fully incorporate the process in our minds.

A source of error could have come from the fact that we ignored the tension in the strings when we applied the conservation of energy equation. This most likely contributed to the ball's calculated initial velocity being smaller than it actually was. Another source of error could have came from the apparatus used the measure the angle. There was probably a little bit of friction at the pivot of the marker. This could have also led to a smaller calculated value versus the actual value.