Video Transcript
The change in the velocity of two objects with time is shown in the graph. Did the two objects have the same speed? Have the two objects been moved equal distances from their initial positions?
Looking at our graph, we see that it shows us the velocity of two objects, we’ll call them the red object and the blue object, against time. We also see these two horizontal dashed lines. And we see that the red object starts at the top dashed line and ends at the bottom one, while the blue object begins at the bottom one and ends at the top line. Along with this, the red and the blue lines both cross the time axis at the same time value.
Now, there’s something interesting we can discover about these two dashed lines. Because our red and blue objects both begin moving at the same time value and then they also end their motion at the same time value and because they cross the time axis at the same point, as we saw, all of this taken together means that the distance, we could say, of our top dashed line from the time axis is the same as the distance from the time axis to the bottom dashed line. In other words, these two lines are the same distance above and below our horizontal axis.
Knowing this, let’s now consider our first question. Do the two objects, the red and the blue ones, have the same speed?
Now our graph doesn’t show us speed versus time, but it does show us velocity versus time. And we can recall that speed and velocity are related. The magnitude of an object’s velocity is equal to its speed 𝑠. From this equation, we can see what we would do mathematically to go from an object’s velocity to its speed. We would take the absolute value of its velocity.
So let’s now apply this idea to our graph. Let’s start with the motion of the red object represented by this red line. We know that this line represents that object’s velocity, but we want to know what its speed is. So we’ll take the absolute value of the red object’s velocity, which means that any values of this velocity which are already positive will stay the same — they’ll stay as they are — but any values that are negative, such as all of these velocity values down here, will become positive. That happens by reflecting this portion of the red line about the time axis.
If we represented that with a dashed line, it would look like this. Remember that we haven’t changed the magnitude of the velocity but only its sign. We’ve changed it from negative to positive so that now it’s a speed. So while the velocity of the red object looks like this versus time, the speed of the red object looks like this. And that’s because, as we saw, speed is always positive, whereas velocity can be negative.
Now let’s think about the motion of the blue object. Just like with the red object we’ll take the absolute value of the blue object’s velocity to solve for its speed. And once again, this means leaving any positive values as they are but reflecting negative velocity values about the time axis. That means that this whole portion of the blue line will be reflected about that horizontal axis. Representing that reflection again with a dashed line, it would look like this.
We’re seeing then that while the velocity of the blue object looks like this over time, the speed of that same object looks like this. And now we’re able to answer this question of whether the two objects have the same speed. Graphically, the speed of the red object over time goes like this, and the speed of the blue object over time goes like this as well; they overlap one another. At any point in time, they both have the same speed. And so we answer yes to this question.
Now, let’s consider question number two. Have these two objects been moved equal distances from their initial positions?
To start figuring this out, we can recall that the speed of an object 𝑠 is equal to the distance it travels divided by the time it takes to travel that distance. This means that if we multiply both sides of this expression by the time 𝑡, that term cancels out on the right. And we see that the distance an object travels is equal to the speed at which it travels that distance multiplied by the time it takes to move that distance.
What we now want to do is figure out whether our red and our blue objects moved equal distances from their initial positions. Since we’re specifically talking about distances here, and not displacements, we’ll want to focus in not on the velocity of our objects but rather on their speeds. So for the purposes of this question, we’ll neglect these negative portions of these red and blue lines. Those portions represent perfectly valid velocities, but again, for this part of the question, we’re focusing on speeds and distances. All of the action as far as that’s concerned takes place for nonnegative values.
Now our equation tells us that if we take the speed of our objects at some particular time value and we multiply that speed by that time, then that tells us the distance the object has traveled. This tells us that if we go to our blue and red lines, the ones representing object speed, then we could solve for the total distance these two objects have moved by multiplying together all the various speed and time corresponding pairs along these two lines.
Graphically, doing this would be equivalent to calculating the area under this part of our speed curve and then adding it to the area under this part of our curve. These two areas combined would represent the total distance traveled by our objects from their initial positions.
Now, we don’t have the exact values we’d need to actually go about calculating this distance, but from our blue and red lines representing the objects’ speeds, we can see that these distances would be the same for each. That is, the area under the red speed curve is the same as the area under the blue speed curve. And therefore, their total distance traveled is the same. This is because those curves overlap one another. And so we can answer yes to the second question as well. The two objects have moved through equal distances from their initial positions.
