The velocity function (in m / s ) is given for a particle...

The velocity function (in $\mathrm{m} / \mathrm{s}$ ) 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.
$$
v(t)=3 t-5, \quad 0 \leqslant t \leqslant 3
$$

Key Concepts

Velocity Function

A velocity function describes the speed and direction of a particle's motion as a function of time. It is the derivative of the position function and provides the primary basis for analyzing motion in one or more dimensions.

Definite Integration

Definite integration involves computing the area under a curve within specified limits. In the context of motion, it is used to accumulate quantities such as displacement and total distance traveled over a time interval by integrating the velocity function or its absolute value.

Displacement

Displacement is the vector representing the overall change in position of a particle over a period of time. It is calculated by integrating the velocity function over the given time interval, which results in a net change that accounts for both forward and backward movements.

Distance Traveled

Distance traveled is a scalar quantity that measures the total length of the path taken by a particle, regardless of direction. It is determined by integrating the absolute value of the velocity function over the time interval, ensuring that all movement contributes positively to the total distance.

Transcript

00:01 Okay, we're given the velocity of a particle, and we want to know the displacement of the particle from zero to three seconds and the distance traveled.

00:12 So if the particle moves in one direction only, then its displacement and its distance are the same.

00:24 Okay, displacement means how far are you from your starting point? so if you only travel in the positive direction, the distance.

00:32 Distance you traveled and how far you are from the starting point are exactly the same.

00:37 Okay, but if you first go forward, stop, and then turn around and go backwards, then the displacement that you've traveled or the displacement of your particle is this much.

00:53 Okay, the change in distance from where they were at the beginning to where it is at the end.

00:59 But the distance traveled is all of this and all of that.

01:06 Okay, so as long as it goes in only one direction, those two things are the same.

01:11 Okay, so if we want to find the displacement of this particle in the first three seconds, then what we have to find is s of 3 minus s of 0.

01:24 And to find s, you just integrate v of t, 0 to 3, 3t minus 5, d t.

01:36 So you get 3 t squared over 2 minus 5t from 0 to 3.

01:46 3 halves times 3 squared minus 15 minus 0.

01:55 So 27 halves, minus 30 halves, minus 3 halves.

02:02 All right.

02:04 So that means if you are, let's say you are on a ruler, if you started at 0, now you're at negative 3 and a half or negative 1 .5 inches.

02:15 Okay, you've gone backwards off the back of the ruler.

02:19 Okay, so that's your displacement.

02:26 Okay, now to figure out the distance traveled, you've got to find out if you stopped somewhere from zero to three seconds and turned around to go the other direction.

02:37 If you don't stop at all between zero and three seconds, then the answer to distance travel and displacement are the same.

02:53 Okay.

02:54 So does this particle stop? well, if it stops, that would be where v of t equals zero.

03:11 So does v of t equal zero? so 3t minus 5 equals 0 at t equals 5 thirds.

03:22 And 5 thirds is between 0 and 3...

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