Question

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

v(t) = 3t − 8, 0 ≤ t ≤ 3
(a) Find the displacement.


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

Answer 1

The displacement of a particle with the given velocity function over the time interval is -10.5 meters, and the distance traveled is 11.5 meters.

Step-by-step explanation:

To find the displacement of a particle with velocity function v(t) = 3t - 8 over the interval 0 ≤ t ≤ 3, we need to integrate the velocity function over the given time interval.

1. ∫0 to 3 (3t - 8) dt = [1.5t^2 - 8t]∫ from 0 to 3

= (1.5 × 9 - 8 × 3) - (0 - 0)

= 13.5 - 24

= -10.5 meters.

This is the displacement.

2. To find the distance traveled, we must consider the magnitude of displacement during each interval where the velocity is positive or negative.

Since v(t) changes sign at t=8/3, we calculate the distance traveled separately for 0 to 8/3 and 8/3 to 3:

• ∫0 to 8/3 (3t - 8) dt = -10.67 meters (negative due to the negative velocity).
• ∫8/3 to 3 (3t - 8) dt = 0.83 meters (positive).

The total distance is the sum of the absolute values: 10.67 + 0.83 = 11.5 meters.