The velocity Function

Question

The velocity function (in meters per second) is given for a particle moving along a line.

v(t) = t² − 2t − 15,     2 ≤ t ≤ 8

Find the displacement (in meters).

Find the distance traveled (in meters) by the particle during the given time interval.

 

Answer 1

displacement = ∫_{(2 to 8)} v(t) dt

distance = ∫_{(2 to 8)} |v(t)| dt

The distance integrates the absolute value of the velocity (called speed), i.e. the direction of motion doesn't matter, the distance traveled always increases.

The absolute value:

|v(t)| = - v(t) from t = 2 to 5, since v(t) is negative there

|v(t)| = + v(t) from t = 5 to 8, since v(t) is positive there

So, you have to split the integral:

distance = ∫_{(2 to 5)} ( - v(t)) dt + ∫_{(5 to 8)} v(t) dt

I assume you can take the simple integrals yourself. I hope that helps.