A Question Of Time

A student is doing an experiment timing a marble rolling down a 45 degree incline plane (see Figure 1 below). A second student says "I’ll bet I can get it to roll to the bottom in less time by using two incline planes like this." (see Figure 2 below). The first student says "Yes your marble will roll faster during the first section, but it also has further to roll. My single plane is the shortest distance between the top and the bottom, therefore my marble will reach the bottom first". Who is right?

• A. The first student is correct. Even thought the second marble rolls faster during it’s trip, it has farther to roll so it takes longer than the first.
• B. The second student is correct. The second marble travels at a faster average speed during its journey, and even though the path is somewhat longer, the faster average speed is great enough that the second marble makes the journey in less time that the first marble.
• C. Neither one is correct. Both marbles take the same time to reach the bottom. The faster average speed of the second marble is exactly offset by the longer distance it has to travel.
• D. This is a hard problem. Why don’t they just throw the marble down?

See if you picked correctly.

Answer

A Question Of Time - Answer = B

Answer: The correct answer is B. The second marble does reach the bottom in less time. (One can solve both cases using the motion equations for constant acceleration and determine the second time is shortest.) This shows that the shortest distance between two points is not always the fastest as far as kinematics is concerned. There is a least time path for each case which does not normally correspond to the least distance path (except in the case of light - which always travels with the same velocity regardless of path. So for light the least distance path is also the least time path.)

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