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!
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..
This scientific problem – well, a more general one – has been solved in the following paper:
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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