Coefficient of Restitution

A bead dropped on the tabletop was observed to rebound to abou 90% of its original height.  A marble dropped on the floor rebounded to an estimated 35% of its original height.

`qx001.  Symbolic Solutions.  Most students will need to work through the details of subsequent specific problems before attempting the symbolic solution.  However if you can get the symbolic solutions, you will be able to use them to answer the subsequent questions.  If you prefer to work through the subsequent questions first, please do.  if you get bogged down on this, move on to the subsequent questions. 

`qx002.  Assume that the bead was dropped from a height of 80 cm and rebounded to 90% of this height.  Analyze the motion on the uniform-acceleration interval of its fall, and then on the uniform-acceleration interval of its subsequent rise.  Assume the net force during each interval to be equal to the bead's weight.

`qx003.  Assume that the marble was dropped from a height of 120 cm and rebounded to 35% of this height.  Analyze the motion on the uniform-acceleration interval of its fall, and then on the uniform-acceleration interval of its subsequent rise.  Assume the net force during each interval to be equal to the bead's weight.

`qx004.  How can you predict the percent of momentum retained from the percent of the original height to which an object rises after being dropped?

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If you feel you have worked out the answers to all or most of the numerical questions correctly, but haven't yet worked out the symbolic solution, you should consider returning to the first question.  However don't get bogged down for a long time on that solution.

Acceleration of toy cars due to friction

The magnitude of the frictional force on a rolling toy car is the product of the coefficient of rolling friction and the weight of the car.

The coefficient of friction can be measured by placing the car on a constant incline and giving it a nudge in the direction down the incline.  It will either speed up, slow down or coast with constant velocity.  If the incline is varied until the car coasts with constant velocity, then the coefficient of friction is equal to the slope of the incline.

As before the symbolic question at the beginning can be attempted before or after the subsequent numerical questions.

`qx005.  Answer the following symbolically:

`qx006.  Give your data for this part of the experiment.

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`qx007.  Show how you used your data to find the slope of the 'constant-velocity' incline.

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`qx008.  The weight of your car is given by the symbolic expression m g, where m is its mass.  What therefore is the expression for the magnitude of the frictional force on the car?

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`qx009.  While the car is coasted along a smooth level surface, the net force on it is equal to the frictional force.  What therefore would be its acceleration?

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`qx010.  You also timed the car as it coasted to rest along the tabletop, after having been given a nudge.  What were the counts and the distances observed for your trials?  Give one trial per line, each line consisting of a count and distance in cm, with the two numbers separated by a comma.

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`qx011.  Based on your data what is the acceleration of your car on a level surface?  In the first line give your result in cm/s^2.  Starting in the second line give a brief but detailed account of how you got your result from your data.

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`qx012.  You have obtained two results for the acceleration of your car along a level surface, one based on the slope of an incline, the other on observed counts and distances.  How well do they compare?  Is there a significant discrepancy?  If so, can you explain possible sources of the discrepancy?

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Interaction between magnets mounted on toy cars

It should be very plausible from your experience that the two magnets exert equal and opposite forces on one another, so that at any instant the two cars are experiencing equal and opposite magnetic forces.  This is not generally the case for frictional forces.  However if frictional forces are considered to have negligible effect while the magnetic forces are doing their work, we can assume that the cars experience equal and opposite forces.

As before you may if you wish save the question of symbolic representations until you have worked the situation through numerically.

`qx013.  When released from rest we observe that car 1, whose coefficient of rolling friction is mu_1, travels distance `ds_1 while car 2, whose coefficient of rolling friction is mu_2, travels distance `ds_2 in the opposite direction. 

`qx014.  When the two cars were released, what were their approximate average distances in cm?  Give your answers in a single line, which should consist of two numbers separated by commas.

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`qx015.  Assuming the accelerations you determined previously for the cars, and assuming that they achieved their initial speeds instantly upon release, what were their initial velocities?

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`qx016.  What was the ratio of the speed attained by the first car to that of the second?  Which car do you therefore think had the greater mass?  What do you think was the ratio of their masses?

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Bungee Cord and Chair

`qx017.  Symbolic solution:  Suppose the average force exerted by the bungee cord on the chair, as it moves between the equilibrium position and position x, has magnitude k/2 * x.

`q018.  When the bungee cord was pulled back twice as far and released, the chair clearly coasted more than twice as far.  Assuming that the average force exerted by the bungee cord was twice as great when it was pulled back to twice the distance, how many times as much energy would the chair be expected to gain when released?

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