If I wrote freshman-year kinematics exams…
…I would definitely include the following problem:
You live in the dorms and your upstairs neighbor, LeBrian Skinner, is a serious basketball player. He is about to declare for the NBA draft, but he fears that his merely average height will put him at a disadvantage. To compensate for his relative shortness, LeBrian decides that he needs to have a vertical jump of at least 36 inches.
In the evening you can hear LeBrian practicing his vertical leap, since he lives directly above you: you hear a loud creak when he first jumps followed by a loud thump when he lands again. You use a stopwatch to time the interval between the moment he first leaves the floor and the moment when he lands again. You measure this interval as 0.8 seconds.
Assuming that LeBrian lands with his legs fully extended (in the same position as when he leaves the floor), how high is he jumping? Is it enough?
For those who are curious, the solution is after the page break.
(sorry for the poor image quality).
Of course, in the real world the answer “31 inches” isn’t enough. You need to have a sense of how accurate that answer is (are you really sure that LeBrian has no chance of getting drafted?) I personally think that all freshman-level physics classes should be required to teach their students enough to answer this second part of the question:
Suppose you only measured the time interval between takeoff and landing of LeBrian’s jump with an accuracy seconds. What is the uncertainty in your calculation of the height of the jump?
The solution to this second part goes like this:
Let be the uncertainty in the timing. Call the uncertainty in the height of the jump.
In part 1 of the question, we derived , where is the full duration of the jump.
The uncertainty in the jump height satisfies .
The derivative .
So in the end, the answer is that LeBrian’s vertical jump is something like inches.
By the way, I’m not sure what this problem says about me. Apparently I’m not arrogant enough to give even my fictional alter ego a great chance of making it to the pro’s, but I’m arrogant enough to say that it’s within uncertainty.