An object on a spring oscillates about its equilibrium position at y=0. Its displacement, f(t), from equilibrium is given as a function of time, t, by f(t)=e^(−t)cos(5t).

Question

Find the greatest distance the object goes away from the equilibrium for t≥0

Answer 1

Hi Michael! Let's try to conceptually understand what we're looking at with this equation. So looking at your equation, we know two things: e^-t means the function exponentially grows smaller as time goes on. cos(5t) means the function waves up and down as time goes on. Putting those two together since they are multiplying together, we know our maximum distance away from equilibrium has to be one of the earliest peaks in the wave. Since we're looking at a cosine function, the first (and highest, for our function) peak has to be when t = 0. This is where we can expect to find the greatest distance, so plugging it in

f(0) = (e^-0)*(cos(5*0)) = (1)*(1) = 1

1 is our greatest displacement!