Posts Tagged ‘physics problems’

Leakier, Slower, and No Rain

December 7, 2010

A while ago, I asked a standard freshman physics problem about a cart that has rain fall into it, then opens a hole and rain leaks out. Then I gave an answer saying that as rain falls vertically into an open cart running on a frictionless track, the cart slows down, but as rain leaks out it shows no change in speed.

That was mostly correct, given the picture I drew of the hole:

leaky cart with hole in the center

Water leaks out a hole in the bottom of the cart as it slides to the right.

The key is that the hole is in the center. Yesterday, Martin Gales posed a question on Physics.StackExchange pointing out that this makes a difference, because if we imagine a stationary cart with a hole all the way to the left, then as it drains, the water moves left, and so the cart will have to move right a little to conserve momentum. But then once the cart starts moving the water leaking out of the cart is moving…

I spent three hours last night trying to solve this seemingly-trivial problem. (My answer is at the original question.) It’s simple enough to pose to first-term freshmen, and yet I went through dozens of slightly-wrong ideas and calculations before hitting on the surprising answer. Further, once I knew what happens, it didn’t seem very complicated any more, leaving me to wonder what the hell is wrong with my overclocked simian brain. The feeling you get when thinking about such a problem is an asymmetric oscillation of healthy frustration and premature joy unparalleled in other pursuits. I want to be mind-fucked like this every night.


New Problem: Leaky, Rainy, and Slow

February 22, 2010

Here’s a classic physics problem I asked my MCAT students yesterday. They unanimously chose the wrong answer.

A cart runs along a frictionless track on a rainy day. The rain falls straight down, and some of it lands in the open cart. As the cart accumulates rain, does it slow down, speed up, or keep going the same rate? (Do not worry about the cart running into raindrops ahead of it. We imagine that the raindrops fall in such a way that they either land in the cart or don’t hit it. Also, there’s no wind resistance.)


Rain falls straight down into the cart, which is coasting to the right.

Next, the rain stops, but the cart gets a leak. Water pours out a hole in the bottom of the cart. Does the cart get faster, slower, or stay the same speed? How does its final speed, when all the water has leaked out, compare to its original speed before the rain? (Again, ignore friction.)

Water leaks out a hole in the bottom of the cart as it slides to the right.

The answer is now up here.

New Problem: Spool of String

December 27, 2008

Five and a half years ago I saw a simpler version of this question on the physics quiz Caltech sends to incoming freshman so they can prove how smart they are before arriving. This generalization appears in the first chapter of Motion Mountain, a free PDF textbook by Christoph Schiller, which purports to cover all of modern physics, integrating the theoretical and empirical viewpoints. I was linked here by a blog post from by Bee at Backreaction.

You have a spool of thread resting on a table, like so.

Picture of spool problem.  There is friction between the spool and table, and gravity is as normal.

Picture of spool problem. There is friction between the spool and table, and gravity is as normal.

You pool on the thread very gently. What is the behavior of the spool as a function of the angle the thread makes with the horizontal? Assume the spool rolls without slipping. You should find two separate domains of behavior (one where to spool rolls forward, one where it rolls backward). What is the critical angle separating them? Provide a geometric interpretation of this result. Or, if you are sufficiently clever (I was not), skip right to the geometric interpretation, and use this to derive the critical angle.

What if the table is inclined with respect to the horizontal?

I only read as far as this first problem, so I don’t know yet whether the book actually provides the answer or simply asks the question. I don’t mind if you cheat, honestly. So long as you think the answer is pretty, however you get it.