an object at the origin at time t = 0 has velocity measured in meters per second,

Question

an object at the origin at time t = 0 has velocity measured in meters per second,

t/20 if 0 ≤ t ≤ 60

v(t) = { 3 if 60 < t ≤ 120

           9-t/20 if 120 < t

When, if ever, does the object return to the origin? t = ____ 

Answer 1

Displacement needs to be zero, so if T is such a time, integral from 0 to T of v(t) has to be zero.

Clearly as v(t) >=0 on the interval [0,120] T > 120

Displacement at T= 120 is integral of (t/20) from 0 to 60 plus 3 x 60 = 90 + 180 = 270

So we need integral from 120 to T of v(t) to be -270

The integral of v(t) when t>120 is 9t - (t^2/40). So you evaluate that between 120 and T then set that equal to -270 and solve for T.

There is a slight issue because v(0) = 0 so can we assume the particle moves away from the origin after t=0 ? I think we have to assume it does.