TBR Physics 3.1 Question 1

[MedPR]

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The question is about water falling from the top of a dam.

The dam is 150m high. If the water strikes the base of the dam with a horizontal speed of 47.2m/s, then what percent of the potential energy at the top of the dam is converted into kinetic energy at the bottom of the dam?

The explanation for this question is way over my head. I got the right answer using a different process. Is there a reason why my method wouldn't work in other situations?

PE = mgh. KE = 1/2mv^2

Cancel out the m since mass of water is the same throughout.

gh=1/2v^2

plugging in numbers

10(150)=1/2(47^2) -------------- 47 is close to 50, and 50^2 is 2500
1500 = 1/2(2500)
1500=1250

1250/1500 reduces down to 5/6 which is 83%. Since you rounded 47 up to 50, your actual answer is a bit less than 83%.

 

[ljc]

I would have approached it nearly the same way as you. I would have figured out potential energy at the top (150*10=1500J), converted that all to kinetic energy (1500=.5v^2, v=~55), then compared the actual value to the theoretical value (47.2/55=~85), similar to comparing theoretical to experimental yield. Our methods are nearly identical, and I can't think of a situation where it'd get you into trouble.

What were the answer choices, and what was their explanation?

Edit: also, you say 47.2m/s horizontal velocity -- shouldn't it be vertical?

 

[MedPR]

I would have approached it nearly the same way as you. I would have figured out potential energy at the top (150*10=1500J), converted that all to kinetic energy (1500=.5v^2, v=~55), then compared the actual value to the theoretical value (47.2/55=~85), similar to comparing theoretical to experimental yield. Our methods are nearly identical, and I can't think of a situation where it'd get you into trouble.

What were the answer choices, and what was their explanation?

Edit: also, you say 47.2m/s horizontal velocity -- shouldn't it be vertical?

The question says 47.2m/s horizontal. I'm pretty weak with energy/work right now so maybe it is a typo and should say vertical.. I don't know. It definitely says horizontal though

The answers were 11.1%, 47.2%, 75.8% and 96.4%. So picking the right one from 85% wasn't hard at all.

Here's the explanation.

The water falls from a height of 150m. If all of the water's initial PE were converted into KE, then we could say that mgh initial = 1/2mvbase^2. The vertical speed starts at 0, so we know the vmaxbase^2 = 2gh. Plugging in we get:

vmaxbase^2 = 2*10*150 = 2*1500 = 3000
Rather than solve for v, we should leave the value in terms of v^2 because KE is proportional to v^2. We now need to solve for the actual v^2 and then determine the ratio of the actual v^2 to the max v^2.

v^2 actual = 47.2^2 = 50^2 = 2500

1/2mvactual^2 / 1/2mvmax^2 = vactual^2 / vmax^2 = 2500/3000 = 5/6 = .833 =83.3%

We know that the 2500value we used was a bit too high, so the actual percentage should be a little less than 83.3%. The closest answer is choice C, which makes it the best answer.
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There's more, but it's just a method of showing you that had you not rounded 47 to 50, you would've gotten 75%.

I guess the answer makes sense now, but why would you do it that way? It seems like too much work (read: takes too long) for an MCAT problem.

 

[MrNeuro]

rather than compare KE they compared velocities (more specifically v^2) because KE is proportion to v^2

i personally did the exact same thing as you and i think thats the "better" path as the question is asking about energy and issues can arise with nonconservative forces if you forget that vmax^2=2gh only works for conservative forces.

what i still don't understand is the whole BINOMIAL EXPANSION explanation do you?

"But for some of us, 83.3% is too far away from 75.8% to feel comfortable. If that's the case and you can do math quickly, there is a nice trick for solving squares. Using binomial expansion, we know that (x+y)(x-y)= x^2-y^2, so 97 x 3 = 50^2 - 47^2...47^2 = 50^2 - (97x3) = 2500-291=2209. The ration of 2250:3000 is 3/4= 75%"

 

[MedPR]

rather than compare KE they compared velocities (more specifically v^2) because KE is proportion to v^2

i personally did the exact same thing as you and i think thats the "better" path as the question is asking about energy and issues can arise with nonconservative forces if you forget that vmax^2=2gh only works for conservative forces.

what i still don't understand is the whole BINOMIAL EXPANSION explanation do you?

Nope. If the answers were so close that exact math was needed, I'd probably be able to do 47*47 faster than figure out that binomial expansion crap. Multiplying long hand doesn't take that long anyway.

Edit: Or I guess do 40^2 = 1600 = 16/25 which is a little more than 3/5 so a bit higher than 60%. If you had multiple answers between 60% and 85% that wouldn't work either, but I don't think MCAT would require you to be much more exact than that.

 

[MT Headed]

I would have approached it nearly the same way as you. I would have figured out potential energy at the top (150*10=1500J), converted that all to kinetic energy (1500=.5v^2, v=~55), then compared the actual value to the theoretical value (47.2/55=~85), similar to comparing theoretical to experimental yield. Our methods are nearly identical, and I can't think of a situation where it'd get you into trouble.

You can't compare vexpected vs. vactual. You have to compare 0.5 m vexpected^2 to 0.5 m vactual^2, or after dividing by common factors, you have to compare vexpected^2 to vactual^2. The mcat would be far more likely to make it easy to calculate both the v ratio and the v^2 ratio, and offer them both as answers to trap the unwary.