PhysicsLAB: Conservation of Momentum in Two-Dimensions

In this lab we will investigate conservation of linear momentum in two-dimensions by allowing a large metal ball bearing to roll down an incline and collide obliquely with a stationary, smaller ball bearing. Afterwards we will calculate the total kinetic energy of both balls before and after the collision to determine what percentage of the energy was lost during the collision.

Equipment Needed

As shown in the above picture, each group will need a ramp, white paper, carbon paper, large ball bearing, small ball bearing, a ruler and a protractor.

Experimental Procedure

Place the carbon paper ("inky side up" under the white target paper. Place the flat portion of the ramp on top of the white paper and use a pen to mark its edges.

Next place the small ball bearing slightly off to the side of the ramp so that the large ball bearing will strike it off-center after rolling down the ramp.

After the ball bearings collide, they will leave tracks on the BACK of your white target paper. Notice that you will be viewing a mirror image, that is, the small ball bearing's track will be on the bottom and the large ball bearing's track will be on the top.

Your next step will be to "connect the dots" for each track. Drawing your lines back until they cross.

Once the intersection has been found, you will use a protractor to draw in the "x-axis" for the collision. this is a line which is perpendicular to the front edge of the ramp which passes through the intersection of the tracks.

Next use a protractor to measure the angle that each track makes to your "x-axis." Read your protractor CAREFULLY to one-decimal place. Record your measurements on your target paper.

Refer to the following information for the next five questions.

Data

How high was the top of the ramp above the top of the table in cm?

What was the mass of the large ball bearing in grams, m_L?

What was the mass of the small ball bearing in grams, m_S?

At what angle did the large ball bearing leave the collision, q _L?

At what angle did the small ball bearing leave the collision, q _S?

Using conservation of energy methods, determine the velocity of the large ball bearing when it reached the bottom of the ramp, prior to its collision with the small ball bearing.

The velocity of the large ball bearing (in m/sec) when it reached the bottom of the ramp, v_{ramp base}, was

Refer to the following information for the next two questions.

Velocity Analysis

In this lab, each of the ball bearings leaves the collision at a unique angle and speed. To determine just how fast each one is moving, we will need to look at the equations for conservation of momentum in both the x- and y-directions.

Since the large ball bearing did NOT have any y-momentum prior to the collision, conservation of momentum tells us that the y-components of the momenta of the ball bearings after the collision must be equal and opposite; that is, they must add to zero.

Since you know the values for both of the ball bearings' masses (m_L and m_S) as well as the angles that their paths made to the x-axis (q_L and q_S) you can simplify this equation to one that only has two unknowns, v_fL and v_fSwith their necessary coefficients. We will refer to that equation as Equation #1.

As far as the x-momenta are concerned, we can write a similar equations using the values for the horizontal components of each ball bearing. Remember that the large ball entered the collision moving completely in the x-direction.

Once again, you know the values for both of the ball bearings' masses (m_L and m_S) as well as the angles that their paths made to the x-axis (q_L and q_S) you can simplify this equation to one that only has two unknowns, v_fL and v_fS and their necessary coefficients. We will refer to this equation as Equation #2.

Now you must use simultaneous equations to solve for v_fL and v_fS. There are two easy methods:

The first method is called the substitution method and involves solving the first equation for v_fL in terms of v_fS and substituting its expression in to the second equation, giving you an equation with only v_fS. Find v_fS. Substituting v_fS back into the first equation, solve for v_fL.
The second method is called the addition-subtraction method and it involves matching the coefficients of either v_fL or v_fS so that you can either add the system to eliminate the variable or subtract the system. Once you know one value, substituting it back into either of the original equations will give you the correct value for the second variable.

What is the value of v_fL in m/sec?

What is the value of v_fS in m/sec?

Refer to the following information for the next five questions.

To finish our analysis, we need to calculate the total kinetic energy before the collision and the total kinetic energy after the collision and see how they compare.

Remember that the total KE before the collision is actually the large ball bearing's potential energy at the top of the ramp.

Recall the both ball bearings are translating as well as rotating across the target paper. So their kinetic energies need to include expressions for both KE_{translational} and KE_rotational.

1. What was the large ball bearing's total kinetic energy before the collision? Express your answer in Joules.

2. What was the large ball bearing's total kinetic energy after the collision? Express your answer in Joules.

3. What was the small ball bearing's total kinetic energy after the collision? Express your answer in Joules.

4. How many Joules of kinetic energy was lost during the collision?

5. What percent of the large ball bearing's original kinetic energy (Question #1) does this loss (Question #4) represent?