Question

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 4 sin πt + 3 cos πt, where t is measured in seconds. (Round your answers to two decimal places)
Find the average velocity during each time period.
[1,2]

Answer 1

Velocity is the derivative of displacement with respect to time.\[s(t)=4sin(\pi t) + 3cos(\pi t)\]\[\frac{d}{dt} s(t)=4\pi cos(\pi t) - 3\pi sin (\pi t)\]The average velocity would (I think) be given by the sum of the velocities at the beginning and end of the period, divided by 2. A period of sin or cos is 2\(\pi\), so we're looking at t=0, 2. \[\frac{d}{dt} s(0)=4\pi cos(\pi (0)) - 3\pi sin (\pi (0))=4\pi\]\[\frac{d}{dt} s(2)=4\pi cos(\pi (2)) - 3\pi sin (\pi (2))=4\pi\]The average is then 4\(\pi\). Does this make sense? Not to me at least, periodic motion like this should have an average velocity of 0 as far as I'm aware, so there's probably a mistake somewhere in there!

Answer 2

I suspect the average velocity is probably meant to be given by one minus the other. Not sure why though. Good luck!