A block is suspended from a spring, pulled down, and released. Th...

Question

A block is suspended from a spring, pulled down, and released. The block's position-versus-time graph is shown in FIGURE P2.38.a. At what times is the velocity zero? At what times is the velocity most positive? Most negative?

Accepted Answer

Hey, everyone in this problem, the figure given below shows the position versus time graph describing the motion of a body moving in a straight line. We're asked to determine the times at which the velocity is most positive and most negative. So we're looking for times now, the figure we're given on the Y axis, we have the position X in meters on the X axis. We have the time t in seconds. The time ranges from zero seconds up to 10 seconds and the position ranges from zero m up to 30 m and the figure is broken into five segments. So every two seconds, we have a different kind of motion being shown in the figure. The answer choices were given, we have four answer choices A through D and each of them have a different time range for the most positive and the most negative velocity. So we wanna find when the velocity is most positive or most negative, how can we relate the velocity to this position versus time graph? Or we're called that the velocity is equal to D X divided by D T. OK. The change in position over change in time, which is just equal to the slope of the position time graph. So if we want to find the most positive and most negative velocity, we want to find the most positive and most negative slope of this graph. Now we're gonna calculate the slope for each of these segments. And then we can compare, can you rate off of just looking at it? It's hard to tell which slope is the most positive. So let's go ahead and calculate for each of these segments. So we have the segment from T equals zero seconds to T equals two seconds. And our graph shows that at T equals zero seconds, we have a position of zero m and at T equals two seconds, we have a position of 10 m. So we want to calculate the slope. We're gonna call this V one, the velocity of this first segment call it slope is rise overrun. In this case, the change in the Y axis, which is gonna be the change in X M in the position over the change in tea. Now, the change in X, the position, right again, we said that we have a final position of 10 m, an initial position of zero m. We have 10 m minus zero m divided by a final time of two seconds minus an initial time of zero seconds. And so we end up with a speed or a velocity of five m per second for that first thing Moving to the 2nd segment, this is gonna be when T is equal to two seconds, up to T is equal to four seconds. And the velocity V two is gonna be the change in X. And when I say delta X here, I don't mean the change in the X axis, I mean the change in the position that is labeled as X divided by delta T, which is a change in the X axis here. Now let's look at our graph in the final position is equal to the initial position. Here. At four seconds, we have a position of 10 m and at two seconds, we have a position of 10 m as well. And so the change in position is going to be zero m, the change in time, four seconds minus two seconds. OK? We have zero divided by two seconds and so we get zero m per second. OK? And we can pick this off right away by looking at that lot horizontal line when we have a horizontal line, recall that we don't have slope. And so the velocity is zero, moving to the next segment we have from T equals four seconds to T equals six seconds. The velocity in this segment is gonna be the change in the position over the change in time And the final position at six seconds is 30 m. The initial position at four seconds is 10 m. So the change in position Is going to be 30 m minus 10 m. The change in time is our final time, six secondss minus our initial time, four seconds. So we get 20 m divided by two seconds for a velocity of 10 m per second. Moving on, we have T equals six seconds, two T equals eight seconds. Our graph shows that the line in this segment is horizontal. Again, we have no change in position, which means that the velocity is going to be zero just like it was in segment two. So V4, the change in position over the change in time, we have no change in position. We have a change in time of two seconds. And so we end up with a velocity of 0m/s. And we're on to our final segment from T equals eight seconds to 10 seconds. And again, our velocity is going to be the change in the position which is on the Y axis divided by the change in time, which is on the X axis. OK. This is like rise over run to calculate the slope. Our graph shows that at T equals 10 seconds, the position is zero m And at T equals eight seconds, the position is at 30 m. So moving back to our equation, The change in position is going to be the final position which is zero m minus the initial position, just 30 m divided by the change in time, 10 seconds -8 seconds. And this is going to give us a velocity of negative 15 m/s. Now, this is the only negative velocity we had. This is the only negative slope. And so when we're answering the question of which is the most negative, OK, we have negative 15 m per second is the most negative velocity. And that occurs at t equals eight seconds to 10 seconds. And if we look at the most positive, we found velocities of five m per second, zero m per 2nd, 10 m per second and zero m per second again. And so the most positive velocity is 10 m per second and that occurred from T equals four seconds to six seconds. Now, if we go back to our answer choices, we can compare what we found and we found that the most positive velocity occurs from T equals four seconds to six seconds and the most negative velocity occurs from T equals eight seconds to 10 seconds, which corresponds with answer choice. D thanks everyone for watching. I hope this video helped see you in the next one.

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A block is suspended from a spring, pulled down, and released. The block's position-versus-time graph is shown in FIGURE P2.38.a. At what times is the velocity zero? At what times is the velocity most positive? Most negative?

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