phy121
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'. **
2.1cm, 2.1cm
1.5cm
+-.05cm
** Five horizontal ranges of uninterrupted large ball, mean and standard deviation and explanation of measuring process: **
27.8, 28.2, 27.8, 28.2, 28.2
28.04, .2191
I measured from the end of the ramp to the edge of the table. This measurement was 1.8cm. From the edge of the table, to the edge of the plywood closest to the table, was another 8.3cm. Adding both those values to the distance from the edge of the piece to where the ball landed on it I was able to determine the horizontal distance.
** 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. **
50.2, 49.8, 49.8, 51.1, 49.4
22.2, 22.7, 23.2
50.06, .6465
22.7, .5
The first line is the range of the second ball from its position. The second line is the larger first ball's ranges from the end of the ramp. The third line is the mean and SD of the second balls range. The fourth is the mean and SD of the first ball's ranges.
The measurements were made using a similar practice to the previous box, however ignoring the 1.8cm from the ramp to the edge of the table for the second ball, using instead .4cm.
** Vertical distance fallen, time required to fall. **
72.4cm
.384sec
The distance was measured from the center of the ball on the rail to the surface of the plywood on the floor. For the time, I simply used `dt = sqrt[`ds / .5a] and it was close to times shown with the timer program, but I assumed the math would be more accurate, even though it ignores air/wind resistance.
** 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. **
73.0208cm/s, 59.1146cm/s, 130.365cm/s
73.5914cm/s, 72.4503cm/s
60.4167cm/s, 57.8125cm/s
132.048cm/s, 128.681cm/s
** 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. **
73.0208cm/s * m1 = p1
73.0208cm/s * m1 = (59.1146cm/s * m1) + (130.365cm/s * m2)
130.365cm/s * m2 = p2
(73.0208cm/s * m1) + (0cm/s * m2) = ptotal before
(59.1146cm/s * m1) + (130.365cm/s * m2) = ptotal after
(73.0208cm/s * m1) + (0cm/s * m2) = (59.1146cm/s * m1) + (130.365cm/s * m2)
** 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. **
(73.0208cm/s * m1) - (59.1146cm/s * m1) = (130.365cm/s * m2)
since (0cm/s * m2) = 0, I removed it from the equasion.
m1 = (130.365cm/s * m2) / 13.9062cm/s
(m1 / m2) = (130.365cm/s / 13.9062cm/s)
(m1 / m2) = 9.3746
This means that m1 is 9.3746 times larger than m2
** Diameters of the 2 balls; volumes of both. **
I have a metric micrometer, so these are the diameters using it:
m1 = 2.54cm, m2 = 1.02cm
8.58025cm^3, .555647cm^3
** How will magnitude and angle of the after-collision velocity of each ball differ if the first ball is higher? **
The second ball will start with a downward acceleration, decreasing its `dt of fall.
The speed will be the same, but x value will be less, and the y value more.
yes, for the second ball, it will have a downward angle to begin with.
** Predicted effect of first ball hitting 'higher' than the second, on the horizontal range of the first ball, and on the second: **
This would not change the range of the first ball I don't think, but I think the second ball will have a lessend range due to having an initial vy.
** 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: **
10.6935
v(m2) / [v(m1) before - v(m1) after]
128.681 / (72.4503 - 60.4167)
** What percent uncertainty in mass ratio is suggested by this result? **
12%
** 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? **
** Optional additional comments and/or questions: **
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Please let me know if you have any questions related to this orientation assignment.