Velocity, acceleration and distance traveled for points on wave

We are told a transverse wave travels to the right along a string. They draw it right over here. Two dots have been painted on the string in the diagrams below. Those dots are labeled P and Q, so that's these dots here. The figure below shows a string at an instant in time. At the instant shown, dot P has maximum displacement, and dot Q has zero displacement from equilibrium.

At each of the dots P and Q, draw an arrow indicating the direction of the instantaneous velocity of that dot. If either dot has zero velocity, write v equals zero next to the dot. So pause this video and see if you can have a go at it on your own. Before I work through it with you, all right now, let's work through this together.

So let's think about the instantaneous velocity of this dot right over here. To help us think about it, let's think about what's going on on either side of this dot. So if we think about this point right over here, what's going to happen a little bit after this moment in time we're looking at? Well, the wave is traveling in that direction, so in a little while, the wave is going to look— in a little moment after that, the wave is going to look something like this. The wave is going to look something like this. So this dot is in the process of moving down.

And what about this dot right over here? Well, this dot, a little bit later, is going to be higher, so it's moving up. And it's this point where we're at our maximum. When we're at our maximum displacement from equilibrium, we are going from moving down to going up right at that moment. So right at this moment of maximum displacement, we have zero velocity. So let me do that in red so you can see it. So here at point P, we would write v is equal to zero.

Now, what about at point Q? Well, we just sketched what's going to happen a moment later. Our transverse wave is going to look like this, and so point Q is going to be up here, so point Q is moving upwards. So if we want to draw its velocity vector, it would look something like that.

Now let's do the next part. The figure below shows the string at the same instant as shown in part A. At each of the dots P and Q, draw an arrow indicating the direction of the instantaneous acceleration of that dot. If either dot has zero acceleration, write a equals zero next to the dot. So pause this video again and see if you can have a go at it.

One way to think about it is all the points around P, and actually all of the points in this area right over here are going to have a net upward acceleration. So even though this point, for example, right over here has a velocity downwards, that downward velocity is becoming less and less and less as you get closer to P, and then it flips around, becomes an upward velocity and becomes more and more upward as we get closer and closer to this point.

So everything in this area right over here has a net upward acceleration. So P would have a net upward acceleration as well. It has zero instantaneous velocity, but we're at that moment, we're going from a downward velocity to an upward velocity, so we have an upward acceleration.

What about at point Q? Well, one interesting thing to appreciate is that everything in this region right over here has an upward acceleration, and everything over here has a downward acceleration, and then everything over here has an upward acceleration. So point Q is at that transition point where we're going from a downward acceleration to an upward acceleration. And so here, right at that point, at this moment, you're going to have zero acceleration. So here we write acceleration is equal to zero.

And then it says the figure below represents the string at time t equals 0. The same instant as shown in part A, where dot P is at its maximum displacement from equilibrium. For simplicity, Q is not shown on the grid below. Draw the string at a later time, time t is equal to capital T divided by four, where capital T is the period of the wave. So pause this video again and see if you could do that.

If you can even sketch it out, if you don't have access to this drawing right here to think about a full period, we can look at a full wavelength of this wave. One way to do it is you start at a point, and then you see, well, how far do you have to go to get to the same point again? So we went up, we go down, where you could say we're at the same point. But here we're going up, and here we're going down, so we want to go back to when we're going back up.

So that is a full wavelength. A full wavelength here is 24 centimeters. So after one-fourth of a period, this wave would have shifted one-fourth of this length. And so one-fourth would be one of these sections right over here. So all I will do is draw the same wave, but instead of starting here, I'm going to start here— one-fourth of a wavelength to the right, but then I'll draw the exact same wave. It's just shifted. So notice it has just shifted one-fourth of a wavelength to the right, and so it will look something like that.

And then if we look at the next parts of the question on your drawing above, draw a dot to indicate the position of point P on the string at time t is equal to the period divided by four and clearly label the dot with the letter P. So let's just remind ourselves of where P was before. P is at this point that's halfway between 12 and 24 centimeters, so that's going to be at 18 centimeters.

So it's right over here. So at this point, P is going to be right over here. So from t is equal to zero to t is equal to the period divided by four, P went from down here up to here. And then last but not least, it says now consider the wave at time t is equal to a full period. Determine the distance traveled, not the displacement, by dot P between the times t equals zero and t is equal to capital T.

So pause this video and see if you can figure it out. So we already saw at time t is equal to zero, P is right over here. And so what's going to happen over a full period? Well, over that time period, P is going to move all the way up to positive 8 centimeters on our y-axis, and then all the way back down over a full period. And so if you ask this displacement, you would say it's zero; you're back to where you started.

But the distance? Well, it would have traveled 8, 16 centimeters up, and then 16 centimeters back down. So the distance traveled would be 32 centimeters, and we're done.