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In this video, we’re talking about graphing velocity.
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Graphing velocity means plotting out the velocity of some object against time.
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And we can do this for an object that is or is not in motion.
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In order to understand how this all works, let’s consider a 𝑣-versus-𝑡, a velocity-versus-time, graph and let’s consider an object whose motion can be displayed this way as a horizontal line.
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Even though we haven’t made any markings on the axes of this graph to indicate different velocities and times, we can still say that whatever the velocity of this object, that velocity is staying the same.
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It has a constant value all throughout this time period.
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In contrast to this, we could also have an object whose motion can be shown this way on a velocity-versus-time graph.
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We still have a straight line, but now the slope or the gradient of this line is positive.
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This means that as time goes on, the velocity of our object is increasing.
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So, its velocity at this point in time, say, is less than its velocity at this point in time, which is less than its velocity at this point, and so on.
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So then, in this instance, we have an object whose velocity is staying constant, while, in this case, the object’s velocity is increasing.
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And it turns out that there’s another way that we can represent this same object motion.
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We can do this using what’s called a displacement-versus-time graph.
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This is because, as we’ll see, there’s a connection between an object’s velocity and its displacement.
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And at this point, let’s recall the definitions of those terms.
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Velocity indicates an object’s speed and its direction.
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Therefore, velocity is a vector.
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While displacement is the straight-line change in an object’s position, displacement is also a vector.
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We can see that the way an object moves is related to the straight-line change in that object’s position.
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This is how velocity and displacement are connected.
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So, if we look back at this velocity-versus-time graph and the fact that velocity is constant here, we can translate this object motion to a displacement-versus-time graph.
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First, we’ll make sure the start point of our two lines matches up, and then we can say that if our object first starts moving at this point in time, at that instant, we can choose its displacement to be zero.
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An object’s initial displacement could in general be positive or negative or zero, but, in this case, we’ll choose it to be zero for simplicity.
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Therefore, we’ll put a dot on the horizontal axis of our displacement-versus-time graph.
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Now, we’ve noted that we don’t know what this exact velocity value is.
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But we do see that it’s above the horizontal axis, and, therefore, we can assume that this velocity is positive.
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And this means that, as time goes on, this positive velocity will translate into a positive displacement on our 𝐷-versus-𝑡 graph.
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So, we could say that after some amount of time has passed, our object’s displacement has increased to this positive value.
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And then, after more time has passed, its displacement has gotten to here and so on until we get to the end of a line that’s graphed on our velocity-versus-time plot.
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When we connect these points, we see what the displacement-versus-time curve would look like for this particular velocity-versus-time graph.
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Because the velocity here is constant as well as positive, our displacement versus time is positive as well, and it’s always increasing.
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But then, what about a velocity-versus-time curve like this, which already has a nonzero slope or gradient to it?
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This also can be represented on a displacement-versus-time curve.
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Once again, this object’s displacement starts at zero.
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But then, instead of increasing at a constant rate like the displacement did over here, in this case, our displacement does increase.
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But it does so at an increasing rate.
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Here’s how we can think of this.
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At the beginning of our velocity-versus-time curve right here, we have a positive value to our object’s velocity.
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Like we saw over here with our constant velocity, this would indicate an object’s displacement that would steadily increase.
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But rather than staying the same, this object’s velocity increases itself at a steady rate.
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And because the object’s velocity is always going up, the object’s displacement not only increases, but it does so at a faster and faster rate.
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The proof for that is that the slope or the gradient of our displacement-versus-time curve is increasing over time.
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That is, this graph is curving upward.
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So then, if we have an object that has a constant positive velocity, that indicates an object’s displacement which is positive and always increasing at the same rate.
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On the other hand, if the velocity of our object itself is increasing, also at a steady rate, then that object’s displacement will go up.
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But it will do so at a higher and higher rate over time.
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Now, taking a look again at our velocity-versus-time curves, we know that these don’t represent all the possible graphs we could have.
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For example, what if an object’s velocity versus time looked like this?
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That is, the object’s velocity is zero all throughout this interval.
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If we think about an object whose velocity is zero, that means it’s not in motion.
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It’s stationary.
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And that means that its displacement doesn’t change in time.
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If the displacement starts out at zero, then it will remain at zero.
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So, in this case, we have a velocity-versus-time and a displacement-versus-time curve, which are the same.
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This only happens when velocity is zero.
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Now, if we look at this velocity-versus-time curve, we see we have a positive constant velocity.
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And if we look at this one that we just drew, we see we have a zero but constant velocity.
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But what if we had a velocity-versus-time graph that showed a constant negative velocity?
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To show what that looks like, let’s clear away our middle graphs, and then we’ll shift the velocity-versus-time curves on the far-right graphs into the ones in the middle.
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And then, we’ll redraw these two graphs on the right so they more easily admit negative values.
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Okay, so like we said, this graph here is of a positive constant velocity, while this one is of a constant zero velocity.
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Let’s say that on this graph to the far right, we have an object moving with a constant negative velocity.
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This is possible because we are talking about velocities which are vectors and therefore can have positive as well as negative values.
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So we have, in this case, a constant and negative velocity of our object.
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Here’s how that translates to our corresponding displacement-versus-time graph.
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First, like before, we can say that our object begins with a displacement of zero.
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It hasn’t moved at all yet, and so its displacement from its start point is still zero.
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But then, our velocity-versus-time curve tells us our object does start to move, but in a negative direction.
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This means our object will also experience displacement in a negative direction.
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As time passes, this constant negative velocity displaces the object farther and farther in the negative direction, until, finally, at the end of its motion, our object ends up here.
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And if we connect these dots, we know how the object’s displacement versus time curve.
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Now that we have these three pairs of graphs for a positive, a zero, and a negative constant velocity, respectively, we can make an observation.
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Notice that for the displacement-versus-time curve here, the slope or gradient of this curve is positive.
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And this corresponds to a positive velocity.
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Likewise, note that the slope of this displacement-versus-time curve is zero.
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And that corresponds to a velocity of zero.
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And lastly, that the slope of this displacement-versus-time curve is negative.
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And that corresponds to a negative constant velocity.
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We now see that it’s possible to start with a velocity-versus-time curve and generate a displacement-versus-time curve from it, like we’ve done here.
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But this process can also be reversed.
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We could start with a displacement-versus-time graph, calculate the slope of that line, and that will tell us the corresponding velocity of our object.
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This is another way that velocity and displacement relate to one another.
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Now, we’ve mentioned that there’re different kinds of velocity-versus-time curves, and, so far, we haven’t yet looked at one of these curves that crosses over the horizontal axis.
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In other words, we haven’t yet looked at a graph that looks like this.
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The velocity starts out positive, then goes to zero, and then becomes negative.
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Now, earlier, we defined this term velocity this way.
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We said that the velocity of an object indicates the speed of that object, that is, the rate at which it’s traveling, and that it also indicates the direction the object is moving in.
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Now, this is in contrast to the speed of an object.
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An object’s speed simply indicates its rate of motion, in other words, what we’ve called its speed under our velocity heading.
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So, what if we were to do this?
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What if we were to plot this velocity-versus-time curve on this speed-versus-time graph?
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How would that curve look transposed, we could say, from velocity to speed?
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Well, the important thing to see is that while velocity indicates an object’s direction, speed does not.
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So, while the velocity of the object we’re considering did reach negative values here, where it’s below the horizontal axis, that’s not possible on our speed-versus-time graph.
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Speeds are always positive or zero, never negative.
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And that goes back to the fact that speed doesn’t tell us about an object’s direction.
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So, we say that however an object moves or doesn’t move, its speed will always be greater than or equal to zero.
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On our corresponding speed-versus-time curve, we would start out at the same value as our velocity-versus-time curve.
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And until we got to zero, these two curves will look the same.
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But then, as we said, because speeds are never negative, we won’t continue on following the 𝑣-versus-𝑡 line.
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Instead, what will happen, and this is quite interesting, is that that line will be reflected about the horizontal axis.
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What we mean by that is that, at every time value, the distance our dashed line is from the horizontal axis is equal to the distance our real speed-versus-time curve is from that same axis.
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It’s just that the speed-versus-time curve has a positive value, whereas the dash line goes into negative territory.
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This difference between speed and velocity comes down to the fact that velocity is a vector quantity while speed is a scalar.
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Speed tells us how fast an object is moving, but not its direction.
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And therefore, speed is always positive or zero, while an object’s velocity does include direction information and therefore can be less than zero.
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Now, velocity and speed aren’t the only vector–scalar pair that we know of.
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We also know about displacement and distance.
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Now, the way these two quantities relate is like this.
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Say that we had an object that traveled this path, starting here and ending here.
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To calculate the distance traveled by that object, we would measure out all the ground covered by the object on its journey.
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That total amount would equal the distance traveled.
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On the other hand, the object’s displacement is equal to the straight-line distance that it travels from start to finish.
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And since displacement is a vector, the direction the object moves is recorded as well.
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So then, what if we switched out our velocity-versus-time and speed-versus-time curves for displacement versus time and then distance versus time?
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And now, it’s our object’s displacement that goes from positive to zero to negative.
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What, we wonder, is the corresponding distance-versus-time curve?
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One way to help us, as we find out, is to imagine a scenario that would generate this displacement-versus-time curve.
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Let’s say that we have an origin point for displacement, and that origin point is right here as an 𝑥.
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And let’s say further that we decide that motion in this direction to the right from that origin point is what we’ll consider positive motion.
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That, therefore, means that motion the opposite way is considered negative.
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Now, if we have an object that starts out here relative to our origin, then the initial displacement of that object is positive, as we see on our displacement-versus-time curve.
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But then, say that our object starts to move toward this origin and eventually reaches it.
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On our graph, that corresponds to moving to this point where the line crosses over the horizontal axis.
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If our object then keeps moving past this origin, it then goes into negative values of displacement.
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And that’s represented by this portion of our graph.
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Now, because the slope of our displacement-versus-time curve is constant, that means the rate at which our object moves, in this case, from right to left, is also constant.
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But anyway, now that we understand how an object could travel to generate a curve like this, we want to translate this curve to our distance-versus-time graph.
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We can start right here where the object begins its motion.
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Now, recalling that the distance an object travels begins at its start point and then counts up as the object begins to move, we can recognize that as our object just begins to move here, the distance it has traveled at this initial instant is zero.
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So, our distance-versus-time curve starts out on the horizontal axis, with our distance traveled being zero.
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But then, as our object moves along, the distance that it has traveled steadily increases.
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The fact that we’re getting closer to what we’ve called the origin doesn’t impact distance because distance is simply an indication of what we could call the total ground covered.
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So, our distance-versus-time curve, as our object moves towards the origin, would look like this.
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Because distance is a scalar quantity, it can’t be negative.
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And so, we see that these values are all either positive or zero.
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And then, as our object journeys on past the origin point into what we’ve called the negative direction, while our displacement goes into negative territory, our distance traveled does not, but continues on in a positive direction.
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At the end of our object’s journey then, it has a negative displacement from what we’ve called the origin point over here.
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While on the other hand, the total distance it has traveled is positive.
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This is how displacement-versus-time and distance-versus-time graphs can compare to one another.
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And note that, as we made this comparison, it was helpful to sketch out a scenario that could lead to the graph that we were given, in this case, displacement versus time.
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Let’s summarize now what we’ve learned about graphing velocity.
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Starting off in this lesson, we saw that, in general, the velocity-versus-time curve representing the motion of an object could either indicate a constant velocity like here, or a velocity that changes in time.
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We then saw that if we were to translate such velocity-versus-time curves into displacement-versus-time curves, then a velocity with a constant positive value leads to a displacement-versus-time graph with a constant positive slope.
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While a velocity with a steadily increasing value over time leads to a displacement-versus-time graph with a steadily increasing slope.
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Along with this, we contrasted velocity-versus-time graphs with speed-versus-time graphs.
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And we saw that while velocity is a vector and can therefore achieve negative values, speed, which is a scalar quantity, is always positive or zero, never negative.
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And lastly, in a similar way, we contrasted displacement, which is a vector, with the distance, which is a scalar.
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And just like with velocity and speed, we saw that while an object’s displacement can be negative, the distance it has traveled is never negative, but is always either zero or positive.
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This is a summary of graphing velocity.