Total distance traveled question

Homework Statement

Lets say we have the velocity equation for a particle

v(t) = at^3 - bt^2 + ct^ - d with t between 0 and 5

So, to find its displacement I have to integrate v(t) from 0 to 5, and I understand why.
But if I want to find the total distance traveled, I must find where t is negative and then i integrate according to that. So, I would have something like an integral of v(t) from 0 to 4 minus an integral of v(t) from 4 to 5. But why do we subtract? Can anybody explain that to me? We also consider moving backwards when we calculate the total distance traveled right?

Thanks

Mark44
Mentor

Homework Statement

Lets say we have the velocity equation for a particle

v(t) = at^3 - bt^2 + ct^ - d with t between 0 and 5

So, to find its displacement I have to integrate v(t) from 0 to 5, and I understand why.
But if I want to find the total distance traveled, I must find where t is negative and then i integrate according to that.
No, your interval for t is [0, 5], so t is never negative. You need to find where v(t) is negative, because that's when the particle is moving backwards.

If you integrate v(t) from 0 to 5 you'll get the displacement, which is the distance between the particle's position at time t = 0 and time t = 5. For example, if the particle started at the origin, then moved right 5 units, then back 7 units, its displacement would be 2 units, but the distance travelled would be 12 units.
So, I would have something like an integral of v(t) from 0 to 4 minus an integral of v(t) from 4 to 5. But why do we subtract? Can anybody explain that to me? We also consider moving backwards when we calculate the total distance traveled right?
Yes.