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

The velocity function (in meters per second) is given for a...

Key Concepts

Absolute Value in Integration

When calculating total distance traveled, the absolute value of the velocity function is used to avoid cancellation effects from portions of the motion in opposite directions. This ensures that every segment of the path is measured positively, reflecting the true total distance covered.

Integration

Integration is a fundamental tool in calculus used to compute the accumulation of quantities, such as area under a curve. In the context of motion, it is employed to find both displacement and distance traveled by summing the contributions of the velocity function over a given interval.

Distance Traveled

Distance traveled is the total length of the path taken by the object, irrespective of the direction. To compute it, one integrates the absolute value of the velocity function over the time interval, ensuring that all contributions are positive, even when the velocity is negative.

Displacement

Displacement refers to the net change in an object’s position over a specified time interval. It is calculated by taking the definite integral of the velocity function over that interval, accounting for the direction of motion, which can result in positive or negative values.

Velocity Function

The velocity function describes how the speed and direction of an object change over time. It is the instantaneous rate of change of the object’s position, often derived as the derivative of the position function with respect to time.

Transcript

00:01 So we have a particle and its velocity is being described as 3t minus 5.

00:08 And so our test today is to find a displacement and a distance from the interval from t equals 0 to 3, all inclusive.

00:19 So first, let's establish what the difference between this displacement and the distance.

00:27 So assuming that this particle is going on a straight line.

00:32 Then at the first instant, it goes to the right, and then it goes to the left, and it goes right a little bit again.

00:42 It goes all the way to the left, and then it goes right again.

00:46 So, and its ending point is going to be right here.

00:50 And it's not going up, but i just don't want to have all the direction all, like, stack up.

00:59 So i'm just doing it separately.

01:01 So the displacement, it cares about where it's ending in relative to where it started.

01:11 So that's going to be the displacement right there.

01:13 That's going to be the displacement.

01:16 Now the distance is going to be how far it has traveled in total.

01:24 So unlike displacement, the distance it cares about how long this is and how long this is.

01:32 And how long this is, and how long that is, and how long that is.

01:36 And then you add it up all together.

01:40 That's going to give you the distance is all.

01:45 So if you notice, when the velocity of that particle is going negative, right, then it's also adding the distance as well.

01:58 So what we're going to do when we add the distance is we're going to find where the velocity, velocity has an x intercept and where the velocity is negative so that we would take the absolute value of that interval of the negative velocity so let's get started let's find what the displacement is so the displacement is pretty straightforward it goes from zero to three of v of t.

02:37 Which is going to be equal to 3 t minus 5 d t we're going to get 3 halves of t squared minus 5t plus c but we don't really we don't really have to care about the c because it's going to be canceled out in the end so first we're going to plug in three we're going to get 207 halves minus 15 which is going to give us negative three halves meters...

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