Your 'conservation of momentum' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Your optional message or comment: **
** Distances from edge of the paper to the two marks made in adjusting the 'tee'. **
6.8 cm, 6.9 cm
0.8 cm
about 5% uncertainty
** Five horizontal ranges of uninterrupted large ball, mean and standard deviation and explanation of measuring process: **
6.5, 6.8, 7.1, 6.9, 6.8
6.82, 0.2168
I got the first line of numbers by rolling the ball down the ramp, and letting it hit the piece of paper located on the floor at the edge of the table which the ramp is set up on. I then used the meterstick rulers to measure how far the ball traveled horizontally from the time it rolled off the ramp to the time it hit the floor.
** Five horizontal ranges observed for the second ball; corresponding first-ball ranges; mean and standard deviation of the second-ball ranges; mean and standard deviation of ranges for the first ball. **
6.9, 6.9, 7.2, 7.1, 6.8
6.6, 6.7, 6.9, 7.0, 6.7
6.98, 0.1643
6.78, 0.1643
I found these measurements by rolling a large ball down a ramp into a target ball, and measuring the place at which each ball hit the ground using the meterstick rulers given in my lab kit.
Your data are not unreasonable, except that the first ball will not travel further after striking the second ball than it did when it rolled off the edge uninterrupted. The results you report seem to indicate that this is the case.
Your results indicate that after colliding the first and second ball hit in practically the same place. Besides the fact that the two balls do not land on top of one another, this will not happen if the second ball is smaller than the first. If the target ball is smaller, then after a center-to-center collision it will travel significantly further in the horizontal direction than the first.
** Vertical distance fallen, time required to fall. **
78 cm
0.43375 sec
Good.
I determined these distances by measuring the height of the edge of the ramp to the floor to which the balls were falling to. I measured the time by releasing the ball from this distance, and using the TIMER program to find the amount of time it took for the ball to hit the ground.
** Velocity of the first ball immediately before collision, the velocity of the first ball after collision and the velocity of the second ball after collision; before-collision velocities of the first ball based on (mean + standard deviation) and (mean - standard deviation) of its uninterrupted ranges; same for the first ball after collision; same for the second ball after collision. **
15.7, 15.6, 16.09
7.0368 and 6.6032 13.64
6.9443 and 6.6157 13.56
7.1443 and 6.8157 13.96
I don't see in your report any horizontal ranges that would lead to velocities of 6 or 7 cm/s; the larger velocities reported here do appear to be consistent with your data.
** First ball momentum before collision; after collision; second ball after collision; total momentum before; total momentum after; momentum conservation equation. All in terms of m1 and m2. **
momentum = m1 * 15.7 cm/s
momentum = m1 * 15.6 cm/s
momentum = m2 * 16.09 cm/s
total momentum before = (m1 * 15.7 cm/s) + (m2 * 0 cm/s)
total momentum after = (m1 * 15.6 cm/s) + (m2 * 16.09 cm/s)
(m1 * 15.7 cm/s) + (m2 * 0 cm/s) = (m1 * 15.6 cm/s) + (m2 * 16.09 cm/s)
** Equation with all terms containing m1 on the left-hand side and all terms containing m2 on the right; equation rearranged so m1 appears by itself on the left-hand side; preceding the equation divided by m2; simplified equation for m1 / m2. **
(m1 * 15.7 cm/s)-(m1 * 15.6 cm/s) = (m2 * 16.09 cm/s) - (0)
m1 = (m2 * 16.09 cm/s) / 0.1 cm/s
m1 / m2 = 16.09 cm/s / 0.1 cm/s
m1 / m2 = 160.9
You have set up the equations and have done the algebra correctly; however your before- and after-collision velocities for the first ball will not be this close together. The ball does slow more than this; see again my questions about your reported horizontal ranges.
The meaning of the ratio: m1 / m2, is that the 1st mass is bigger than the second mass
** Diameters of the 2 balls; volumes of both. **
1.5, 0.7
1.767, 0.1796
** How will magnitude and angle of the after-collision velocity of each ball differ if the first ball is higher? **
Immediately after the colision, the first ball will decrease in magnitude and will be shot to the ground sooner, while the second ball will increase in inital speed and magnitude, and will travel farther before hitting the ground. If the centers are at the same height, the speeds would be more equally spread between the two balls. The direction would differ slightly, in that the two balls would be more at the same distance.
** Predicted effect of first ball hitting 'higher' than the second, on the horizontal range of the first ball, and on the second: **
the first ball will be sent to teh ground before the second ball, because a lot of the magnitude and speed ofthe first ball is transfered into the second.
** ratio of masses using minimum before-collision velocity for the first ball, maximum after-collision velocity for the first ball, minimum after-collision velocity of the second: **
6.6032 : 6.9443 : 7.1443
-- im not real sure how to find a ratio to put with these values.
** What percent uncertainty in mass ratio is suggested by this result? **
??
** What combination of before-and after-collision velocities gives you the maximum, and what combination gives you the minimum result for the mass ratio? **
?
** In symbols, what mass ratio is indicated if the before-collision velocity of ball 1 is v1, its after-collision velocity u1 and the after-collision velocity of the 'target' ball is u2? **
?
** Derivative of expression for m1/m2 with respect to v1. **
** If the range of the uninterrupted first ball changes by an amount equal to the standard deviation, then how much does the predicted value of v1 change? If v1 changes by this amount, then by how much would the predicted mass ratio change? **
** Complete summary and comparison with previous results, with second ball 2 mm lower than before. **
** Vertical drop of the second ball, its mean horizontal range and the slope of the line segment connecting the two centers; the velocity given by the program based on mean; velocity interval for 2-mm decrease in 2d-ball height; velocity interval from the original run at equal heights; difference in the mean-based velocities; is new velocity significantly different than original? **
** Your report comparing first-ball velocities from the two setups: **
** Uncertainty in relative heights, in mm: **
** Based on the results you have obtained to this point, argue for or against the hypothesis that the uncertainty in the relative heights of the balls was a significant factor in the first setup. **
** How long did it take you to complete this experiment? **
1 hour
** Optional additional comments and/or questions: **
Your analysis is mostly very good; however there are some questions about your data. Don't redo your analysis, but please answer my questions about your data.