How do I solve this?

Question

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

v(t) = 3t − 7,    0 ≤ t ≤ 4

Find the displacement (in meters). I found the displacement to be -4 meters. How do I find the total distance traveled (in meters) by the particle during the given time interval?

Answer 1

Yes, the displacement is -4 meters which you found by integrating.

To find the total distance consider that the initial velocity (at t = 0) is -7 m/s and the final velocity (at t = 4) is +5 m/s. So initially, the particle is moving backwards then it turns around and moves forward. That means at some point the velocity is zero. To find that point make 3t - 7 equal to zero:

3t - 7 = 0

3t = 7

t = 7/3

So the total distance is the absolute value of the displacement between t = 0 and t = 7/3 (since thats the part that's negative) added to the distance traveled between t = 7/3 and t = 4. So break the integral into two parts.

If you are doing the integration using a calculator, you can just do ₀∫⁴|3t - 7| dt

You should get 37/3 meters