## Answer: Leaky, Rainy, and Slow

The problem asked what would happen if rain fell straight down into an open cart as it slid along a frictionless track. Would the cart slow down, speed up, or stay the same speed? Then, if the rain stopped but the cart started leaking, would it slow down, speed up, or stay the same speed?

The answer is that the cart would slow down as rain fell in, and stay the same speed as water leaked out. At the end of the process, the cart would go down the track more slowly than before.

I stated that my MCAT students all chose the wrong answer. They thought the cart would speed up as water leaked out. This is probably because I set them up. I defined momentum, talked about conservation of momentum, and then asked this question.

The question illustrates that momentum is conserved for an entire system, not for individual objects. My students answered the question thinking that the momentum of the cart must be conserved, so that when it gains mass as water falls in, it slows down to keeps its momentum constant. When the cart loses mass as water falls out, they reasoned it would speed up, again keeping its momentum constant.

This reasoning fails because it ignores the momentum of the water. The conserved quantity is the total momentum of the rain and the cart, not just the momentum of the cart.

As the rain falls into the cart, it gains momentum in the direction of the cart’s motion because it gets carried along with the cart. The cart itself (sans water) must lose an equal and opposite amount of momentum. The cart slows down.

As the rain leaks out of the cart its momentum in the direction of motion of the cart doesn’t change. If you were to watch a water droplet leaking out of the cart, it would stay directly underneath the hole as it fell, indicating its momentum in the direction of the cart’s motion is the same as it would have been sitting in the cart the whole time. Since the rain’s momentum in that direction isn’t changing, neither is the cart’s. This keeps the total momentum constant.

This example can be puzzling because it seems we’re cheating somehow. Isn’t rain falling into the cart just the opposite of rain leaking out? So shouldn’t they have the opposite effect?

The difference is that in the first scenario the rain and the cart are moving at different horizontal speeds, but in the second scenario the leaking water and the cart are moving at the same horizontal speeds, so the two situations aren’t opposites of each other and don’t require the cart to act in opposite ways.

Next, one might wonder if this question is at all fair. Would a real cart slow down significantly more in the rain than without the rain?

Very heavy rainfall might be 10 cm/hour of rain, or about 30 grams of rain per second for each square meter of the cart. Let’s imagine a lightweight, plastic cart. I had a sort of pool-type thing shaped like a turtle when I was a kid (until I peed in it too much and my Dad took it away). That probably weighed 10 kilograms and had one square meter of surface area. A cart would need more structure, so let’s say it weighs twice as much, for a total of 20,000 grams per square meter. The cart would lose half its original speed in 20,000/30 = 700 seconds, or about 10 minutes. From experience coasting in a car or my bicycle, it’s very difficult to go near ten minutes losing only half your speed on flat ground. (This depends somewhat on how fast you’re going, since at high speeds you’ll lose half your speed in less time, but even at slow speeds it’s unlikely.) I conclude that the slowdown due to rain is likely much less than the other, ordinary forms of slowdown, and the extra slowdown from rain would be hard to detect.

Additionally, I estimated that the impact of rain on the front of the cart, were we to account for it, would be an order of magnitude greater force than wind resistance (depending on how hard it’s raining and how aerodynamic the cart), so all told this problem must be treated conceptually. We could illustrate it by standing on a ladder and dumping water into the cart all at once as it rolled by underneath, but waiting until it rained would probably lead to a disappointing experiment.