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.
$$
v(t)=t^{2}-2 t-3, \quad 2 \leqslant t \leqslant 4
$$

Key Concepts

Velocity Function

A velocity function represents the rate of change of displacement with respect to time. It shows both the speed and direction of a moving particle, and its sign indicates the direction of movement.

Displacement

Displacement is the net change in position of a particle over a given time interval. It is found by integrating the velocity function over that interval, which accounts for both forward and backward motions by considering the sign of the velocity.

Distance Traveled

Distance traveled is the total length of the path covered by the particle, regardless of direction. It is determined by integrating the absolute value of the velocity function over the interval, ensuring that all segments of the path contribute positively to the total distance.

Transcript

00:01 We're given the velocity of a particle in meters per second, and we want to know what's the displacement and what is the distance traveled in the time from 2 seconds to 4 seconds.

00:15 Okay, so to find the distance traveled, oh, sorry, to find the displacement.

00:25 Okay, we want to do it first because it's the easier one.

00:31 We just need to know where was it at 4 seconds minus where was it at 2 seconds.

00:37 So s of 4 minus s of 2.

00:42 And to find s, which is the position, we just have to integrate the velocity from 2 to 4 dt.

00:54 So 2 to 4, t squared minus 2t minus 3 d t.

01:02 That's t cubed over 3, minus 2 t squared over 2, minus 3t squared over 2, minus 3 t squared over 2, minus 3 t, from 2 to 4.

01:17 So that is one -third t cubed minus t squared minus 3t from 2 to 4.

01:25 So that's 64 thirds, minus 16, minus 12, minus 24 -8 -8 -thirds, minus 4, minus 6.

01:38 So 64 -thirds minus 8 -thirds, that'd be 56 thirds, minus 28 -thirds, minus 28 -thirds, minus 253rds, minus 28, minus negative of 10.

01:53 So 56 thirds minus 18.

02:01 56 thirds minus 30.

02:08 54 thirds.

02:12 Two -thirds.

02:15 So from time two -thirds to time four, we don't know how far they traveled, but we know that they ended up only two -thirds of a meter away from where they started.

02:26 Okay, that's what displacement is.

02:28 So let's suppose they were there at time two and they were here at time four, then this distance between them is two -thirds.

02:39 Okay, now, if they only moved in the positive direction or it only moved in the negative direction, then the distance traveled will also be two -thirds.

02:51 Okay, but if they change directions one or more times, then the distance traveled is not the same because they turn.

03:03 Travel this much, this much, this much, this much, this much.

03:08 Okay, so we've got to see if they turned around at all.

03:12 Okay? did this particle stop ever? did the particle stop? well, if it stopped, that means its velocity was zero.

03:29 Does velocity equals zero anywhere? so t squared minus t minus, oops, t squared minus two, t minus 3 equals 0.

03:45 That's t minus 3, t plus 1.

03:49 So at t equals 3 and t equals minus 1, well, that doesn't make any sense because this is time.

03:57 Oh, t equals 3, it's right in between here.

04:01 All right, so we'll get to decide whether it's going in the positive or negative way...

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