Question

The velocity function (in meters per second) is given for a particle moving along a line. v(t) = 5t − 8, 0 ≤ t ≤ 3

(a) find the displacement (in meters).
____ m
(b) find the total distance traveled (in meters) by the particle during the given time interval.
____m

Answer 1

Displacement and total distance traveled are calculated by integrating the velocity function over time and considering the direction of travel respectively.

Step-by-step explanation:

The student's question asks for the displacement and total distance traveled by a particle with a velocity function v(t) = 5t − 8 within the time interval 0 ≤ t ≤ 3. To find the displacement, we integrate the velocity function over the given time interval. Since the velocity is linear, the displacement can be found by calculating the area under the velocity-time graph, which is a trapezoid in this case.

To calculate the total distance, we need to consider the particle's direction of travel. If the velocity changes sign within the interval, the particle changes direction, and we should add up the absolute values of the areas under and above the time-axis separately. This will provide us with the total distance traveled without regard to direction.