The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval.
v(t) = 3t - 5, 0 ≤ t ≤ 3
(a) the displacement of the particle on [0,3] is equal to the anti derivative of v(t) ,V(t), is equal to
. By finding V(3) - V(0), we can get the displacement of the particle which is -3/2 meters.
(b) To find the total distance the particle has traveled, you need to find out where the particle changes direction. This is when the velocity changing from pos to negative or vise versa. You can find where this occurs by setting v(t) equal to 0. Thus, 0=3t-5. By solving this, we get t=5/3. By testing numbers around this point, we can find out where the velocity of the particle is neg and pos. v(0)=3(0)-5 = -5= a negative number. v(3)= 3(3)-5 = 4 = positive number. From this, we know that to the right of t=5/3, the velocity is pos and to the left, the velocity is neg. That means that at time t=5/3, the particle changes direction.
The the particle traveled, is as follows:
simplifying that, we get the total distance that the particle travels as 41/6 meters