A stuck roller coaster is likely held in place by friction. From the looks of it I would guess a 2 pound weight added to the front of a stuck roller coaster wouldn't do much to budge it, but a 200 pound man hanging off the front seems like more than enough to get it rolling again. As such, I'll assume a 20 pound force is needed start a stopped roller coaster. This force is equivalent to the frictional force holding it in place.
There's a narrow range of heights near the peak at which the cars could stop and not accelerate because friction would be too strong. It looks like the radius of curvature for the track is about 10 meters give or take. The range of heights over which the carts would get stuck is only a small fraction of this. I'll guess 5 centimeters since it's likely bigger than 5 millimeter and smaller than half a meter.
The initial velocity of the cart will correlate with how far it makes it up the hill. Carts with speeds greater than some cutoff velocity will all make it over the hill. Carts with speeds less than some other cutoff velocity will all roll back down the hill. Carts with speeds between these two cutoffs will get stuck. Using conservation of energy (perhaps a dubious move since we're relying on friction to stop us) we can relate the initial velocity to the height of the cart will climb up the hill:
m v2 / 2 = m g h
Here, m is the mass of the carts, g is the acceleration of gravity, and h is the height climbed. The roller coaster might be 100 meters tall. Carts stopping at heights anywhere between 99.95 meters and 100 meters will get stuck. Using the energy equation, we can solve for the range of initial velocities for which this will happen. Leaving out the messy math, you'll find that this occurs for velocities between 70.009 mph and 70.025 mph.
As you can see, there's a narrow range of velocities over which the roller coaster will get stuck. What causes fluctuations in the initial velocity? If we assume each ride gets pushed with the same impulse, then the only fluctuating variable would be the mass of the riders. A roller coaster full of fatties is less likely to make to make it to the top. I'll assume the average person weighs 170 pounds with a standard deviation of about 40 pounds. If there are 50 riders, the standard deviation in weight would be about 3% of the mean total weight. I'll assume the deviation in the initial velocity is the same as that of the weight, i.e. 3% of the mean.1
We still need to find the mean velocity. For the sake of keeping their customers happy, I'm going to assume that roll backs happen only 5% percent of the time. We can assume a normal distribution to find the mean for which this occurs. Skipping the messy details once more, I get about 73.6 mph.
Using this mean and standard deviation, we can then find the fraction of times a cart will get stuck at the top of the hill. Again assuming a normal distribution, I get about 0.1% of the time, which seems like a fairly reasonable estimate.
If the roller coaster operates 8 hours per day and each ride take 2 minutes, you'd have roughly 240 rides per day. Over a 100 day season, you'd have 24,000 rides. Roughly 24 of these would get stuck at the top.
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Aaron Santos is a physicist and author of the books How Many Licks? Or How to Estimate Damn Near Anything and Ballparking: Practical Math for Impractical Sports Questions. Follow him on Twitter at @aarontsantos.
[1] I'm assuming the carts are of negligible weight, which is probably stupid. I suspect it won't throw the number off too much as long as it's comparable to the weight of people.
