Question

What's the definition of linear velocity? From what I know, I think it's when an object has straight line motion, which I think means that there's no directional change. Thus I would think that the motion is 1-dimensional.

Answer 1

The body does not need to be moving in a straight line. The point of linear velocity is to distinguish from angular velocity.

Answer 2

It will essentially be a vector, and each orthogonal dimension will have it's own speed.

Answer 3

Obviously the speed may be changing over time. So you care about the linear velocity at particular times.

Answer 4

Consider a point moving around a circle. We describe its linear position with the vector \[
\mathbf s(t) = \langle \cos(t),\sin(t)\rangle
\]Now it will take \(t=2\pi\) until it ends up at the same place again... doing a full rotation, so we can say angular position is: \[
\alpha(t) = t
\]To get the linear velocity, we take the derivative of the position:\[
\mathbf v(t) =\mathbf s'(t) = \langle -\sin(t),\cos(t)\rangle
\]To get angular velocity, we take the derivative of the angular position \[
\omega(t) =\alpha'(t) = 1
\]Since it is spinning around the circle at a constant rate, this shouldn't surprise us.

Answer 5

You didn't answer my question at all

Answer 6

Oh, sorry. Didn't know I was talking to a fifth grader.

`From what I know, I think it's when an object has straight line motion`
No.

`I think means that there's no directional change`
No.

`Thus I would think that the motion is 1-dimensional`
No.

Answer 7

If it's not straight line motion, then what type of motion do objects with purely linear velocity have?

Answer 8

I know what velocity in general is. There are two general types, average and instantaneous. Average velocity = delta X / delta t and is the slope of the secant line on an x vs t plot. On the other hand, Instantaneous velocity = lim as delta t goes to 0 of average velocity = dx / dt and is the slope of the tangent line on an x vs t plot. Since displacement is a vector, so is velocity.

Answer 9

I'm just confused when the terms "linear" and "angular" velocity come into play. I understand what they are mathematically (purely linear velocity = delta x / delta t or dx / dt whereas angular velocity = delta theta / delta t or dtheta/dt, and v = omega x radius is the equation that connects linear and angular velocity). However, I don't understand them in a less quantiative sense

Answer 10

Linear velocity and angular velocity are both just properties of a physical body. All objects have a linear velocity and an angular velocity (about some axis). Linear is only ever used to distinguish it from angular. A body does not need to be moving at all to have a linear velocity, and it certainly doesn't have to move in a straight line.

Even if a body happens to be rotating, it will still have a linear velocity. Letting the center of mass be the representative for its position, the linear velocity is just the derivative of that position with respect to time.

Thus, so long as a body has a position, it will have a linear velocity. Likewise, so long as a body is associated with some angular displacement about some axis, it will have an angular velocity.