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I was driving uphill from a complete stop for a distance of .4 miles estimated to take 1 minute in a navigation app. I was pulled over right after cresting the hill. The cop had me on radar going 53mph. What do I need to know to see if it was possible to be going that fast uphill ? The grade of the incline? The 0 to 60 time capability of my car? I appreciate any help I can get!

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Is this a word problem or are you in actual legal trouble? –  Michael Brown Mar 26 '13 at 7:44
    
Based on the information you gave, you were travelling at 47.8mph at the top of the hill, not 53mph. –  Mew Mar 26 '13 at 8:05
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Huh? Q: "What do I need to know to see if it was possible to be going that fast uphill ?" A: "The cop had me on radar going 53mph." –  Glen The Udderboat Mar 26 '13 at 9:27
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I don't understand what you expect to get out of this? Are you assuming that your estimates are perfect? Are you assuming a fixed speed the entire time? I mean it seems perfectly possible for you to have been travelling slightly under some fixed speed for the majority of the trip and then sped up at the end... –  Dason Mar 26 '13 at 15:36
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The measurement of 53 mph by radar is most likely much closer to your actual speed than any ex post facto estimate you could calculate. Even if the radar hasn't been calibrated in awhile, the speed that it recorded seems entirely plausible given your description of the situation. Pay the ticket, move on. –  OSE Mar 26 '13 at 16:01
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2 Answers 2

The best way to solve it would be experimentally, by doing the run several times, with calibrated instrumentation by the roadside to measure your speed. The acceleration won't have been constant, so that's not an assumption we can use.

Knowing the 0-60 time capability won't really help; it could be different when accelerating up hill, compared to on the flat. You'll get a boost at the very crest of the hill, as your car goes from accelerating up hill to accelerating on the flat. So the measured speed could be very sensitive to the place at which your speed is measured.

Acceleration won't have been uniform: all you know is that the second integral of the acceleration is 0.4 miles.

Your 1-minute estimate of time taken isn't anywhere near precise enough to be useful - what's the uncertainty on that, $\pm25\% $ ? And the distance estimate of 0.4 miles, assuming 1 s.f., would appear to have an uncertainty of $\pm13\%$.

All in all, there are so many complicating factors, that you will need to run the experiment to get to the bottom of it. This could turn out to be an expensive question to answer.

The question does illustrate the importance of assessing uncertainty in your inputs, and questioning assumptions (such as uniform acceleration) that would make calculation easy, but are unrealistic, and thus would give an invalid answer..

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+1 for being an actual answer –  Michael Brown Mar 26 '13 at 11:18
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Of course, running several experiments might just get you several more tickets. –  Beska Mar 26 '13 at 12:18
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Am I missing something? "The best way to solve it would be experimentally [...] with calibrated instrumentation by the roadside to measure your speed." This experiment has been done. Outcome: $53$ mph (and a ticket). –  Glen The Udderboat Mar 26 '13 at 13:21
    
@Gugg We don't know that the radar is correctly calibrated. If it's not laci has a decent chance of getting the ticket reduced/tossed at court; but since many people find it easier to pay than to take the time off for court some police depts are sloppy about keeping gear in cal. –  Dan Neely Mar 26 '13 at 13:42
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This scientific problem – well, a more general one – has been solved in the following paper:

http://arxiv.org/abs/1204.0162

Because it's legal in my country to move backwards in time, I remember the future event – one minute from now – in which Andrew Gibson will mention that he has this paper hanging in his physics lounge. He will curse me. 11 minutes later, Dilaton will notice that my answer was superluminal, another future event that I remember.

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We have that hanging in my physics lounge, and I was just about to post it. Curse you! –  Andrew Gibson Mar 26 '13 at 8:17
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@AndrewGibson Lumo would never admit it, but he has posted his ansbwer FTL and now he has a problem with a very annoying speeding ticket himself :-P –  Dilaton Mar 26 '13 at 8:28
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Lubos, you may want to quote part of the paper or summarize it. The answer is funny :) , but it's really not more than a link. (that's probably the reason for the downvotes as well) –  Manishearth Mar 26 '13 at 11:11
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I remember reading about this. The professor wrote this paper to get out of an actual speeding ticket, and presented it in court. It worked; he avoided the ticket. Afterwards, he admitted there was an error with his reasoning, and asked if anyone could figure out what it was. [ Edit ]: Linky linky –  BlueRaja - Danny Pflughoeft Mar 26 '13 at 15:44
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Oh of course, the humorless people who are allergic to any kind of harmless fun and downvote good and funny answer use flags because they dont like it when being told that their doing is very uncool, I see :-D. It is a good answer, there is NO reason to downvote, period! –  Dilaton Mar 26 '13 at 16:55
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