Physics Quiz - Solution

A bag filled with sand, weighing m2 = 2 kg, is hanging freely, connected to a nail attached to the wall with a massive rope of a length of l = 0.1 m. At first it is not moving. Now someone throws a tennis ball weighing m1 = 0.1 kg with an average speed of v = 21 m/s directly to its center of gravity. The bag thus accelerates while the ball decelerates until both of them are moving at the same speed v'. Assume that the acceleration of gravity g = 10 m/s^2. What's the angle alpha by which the rope has rotated (see sketch)?

This was a task in an exam in medical physics at the University of Vienna, an obligatory exam which students of human medicine have to take (usually in their first year).

Official solution:

First we have to calculate the speed v'. According to the principle of the conservation of momentum, m1 * v = (m1 + m2) * v'. Therefore v' = v * m1 / (m1 + m2) = 21 m/s * 0.1 kg / (0.1 kg + 2 kg) = 1 m/s.

Next we have to compute the maximum height h to which the bag is elevated. The process of elevation is a transformation of kinetic to potential energy. According to the principle of the conservation of energy, the kinetic energy is equal to the potential energy: m2 * (v')^2 / 2 = m2 * g * h => h = (v')^2 / (2 * g) = (1 m/s)^2 / (2 * 10 m/s^2) = 0.05 m.

Finally, as you can see from the sketch, alpha = arccos ((l - h) / l) = arccos (1 - h / l) = arccos (1 - 0.05 / 0.1) = 60°.

The solution has been found by Peter Byrne, iliks/hugi, UT, Boiled Brain, Tor G. J. Myklebust, Lawrence E. Boothby, William Swanson. Congratulations!

Comments:

There has been some confusion concerning the word "massive". Quoting Lawrence E. Boothby: "The rope is described as 'massive' but the mass is not specified. Given the extreme shortness of the rope relative to the size of the sand bag, the rope is untypical of pendulum problems which ignore the stifness and mass of the rope and assume that the arc is small relative to the length of the rope."

The mass of the rope can be ignored (after all it isn't even specified). I inserted the word "massive" in my translation only to emphasize that the rope wouldn't break. Perhaps I should have skipped it, or I should have used another word instead. I apologize for this source of confusion.

There has also been confusion over the type of the collision.
Peter Byrne wrote: "If the rope is massive the center of gravity of the system depends on the mass of the rope (assuming thats what you meant by 'its center of gravity'). If that isn't what you meant (and you meant the center of gravity of the bag of sand itself) then the question makes even less sense, because the tennis ball and bag will collide either elastically (in which case their velocities will even out and they will mesh/coalesce) but the energy of the collision will have gone into heating sound etc, the rope won't have actually moved yet (which is what I'm assuming whoever came up with the problem thought happened), and they will then move on together at the same velocity (so technically the correct answer to the quiz is 0 degrees). The other possible collision is a partially or wholly inelastic one, I'm assuming that this is not what is being asked, because it makes no sense at all."
Lawrence E. Boothby: "The final speeds of the sand bag and the tennis ball after collision are specified as the same - this would not be the case in an elastic collision where both system momentum and system kinetic energy would be conserved leading to two simultaneous equations which would result in final velocities which were not the same for two bodies."

By definition, in an elastic collision both momentum and energy are conserved, while in an inelastic one only momentum is conserved; some of the energy is converted into heating sound. If the collision were elastic, there would be two equations following from the principles of the conservation of energy and momentum:

(I) m1 * v = m1 * v1' + m2 * v2'
(II) m1 * v^2 = m1 * v1'^2 + m2 * v2'^2

This would result in two different post-collision velocities v1' and v2'. By contrast, if the collision is inelastic, some of the kinetic energy is transformed into heating sound. The second equation becomes:

(IIa) m1 * v^2 = m1 * v1'^2 + m2 * v2'^2 + Q

This enables us to assume that v1' = v2'. Then the task can be solved as described above. So the type of collision in this task is inelastic.

Furthermore, Lawrence made an interesting objection: "In a typical pendulum problem the h sought would be solved by assuming that the kinetic energy of the bag after collision had been completely converted into potential energy at the maximum arc position. If the h desired in your problem is at the point where the bag and ball have the same velocity, then the collision has not been completed yet and some potential energy would be stored in the deformation of the tennis ball and the arc would not have reached its maximum swing yet."

All in all, it seems like this task has some weak points and isn't clear. As far as I know, this task no longer appears in exams in medical physics at the University of Vienna.

Adok/Hugi