Video: Instantaneous Speed and Velocity

In this video we learn about instantaneous velocity and speed, and compare them to average velocity and speed. We learn to use a derivative with respect to time to compute instantaneous velocity.

11:51

Video Transcript

In this video, we’re going to learn about instantaneous speed and velocity. What these two terms mean, why they’re instantaneous, and how to use them in solving problems. To introduce this topic, let’s imagine for a moment that you design roller coasters for amusement parks. And you’re working on a new design for a roller coaster to be installed this coming year. Designing roller coasters is a highly technical task. And with this project, you’re working under a particular constraint. That the maximum velocity the people on the coasters are allowed to experience at any one point is 100 kilometers per hour. With this as the upper velocity limit, you want to find out if there’s any point along the roller coaster’s path where the velocity exceeds this value. And if so, the design will have to be modified. How can you figure out if, under the current design, the coaster’s velocity ever exceeds this maximum upper limit?

To find out, let’s look into these ideas of instantaneous velocity and instantaneous speed. As a bit of background, let’s look at a position versus time plot. On this plot, we have position in meters on our vertical axis and time in seconds on our horizontal axis. And we have the position over time of some object plotted out on this chart. Let’s say that we pick a time somewhere around the halfway mark of the object’s journey. And we say that we’d like to know the velocity of the object at that time. There are a few ways we could approach solving for that particular velocity. One way is to consider that point as the midpoint, roughly, between the start and the end of our object’s journey. We could write that our velocity at the midpoint of our journey is approximately equal to our object’s position at the final time minus the object’s position at the initial time of the journey divided by the final time minus the initial time.

You may recognize this equation as an expression of average velocity. But if we calculate that average, we can see that it might be fairly different from the actual instantaneous velocity, 𝑣, at the point we’re interested in. Seeing that difference, we might think, “well, let’s move our time points of final and initial closer in to our actual time we’re interested in.” Let’s say we move them in closer so that now the slope between them, that is, the average velocity between those two points, more closely matches the instantaneous velocity at the point we want to find it. But as we look at this setup, we see we could still do better. We could move our time points in even closer.

Now we’d move our 𝑡 initial and 𝑡 final until they’re right up against the point we’re interested in. We’re still calculating an average velocity. But that average is approaching more and more our instantaneous velocity. This approximation is best of all so far. Yet, it’s still not exact. Here is what we’ve learned so far. The average velocity of our object is equal to its change in position over the change in time. But if we want to compute an instantaneous velocity, that is, a velocity at a particular time value. That change in time has to approach zero. It’s only as our initial and final times approach one another and indeed become infinitely close. That we’re truly calculating an instantaneous velocity. We can use a special mathematical notation to express this idea. We can say that an object’s instantaneous velocity, that is, its velocity at a particular time value, is equal to the change in its position divided by the change in its time. As that change in time, Δ𝑡, gets smaller and smaller approaching zero.

There’s another way to express this idea using what’s called differential notation. We could say that instantaneous velocity is equal to 𝑑𝑥, where 𝑥 is position, divided by 𝑑𝑡. Built into this notation is the idea that 𝑑𝑡 is infinitesimally small in this expression. Consider for a moment the similarities between average velocity and instantaneous. If we take the form of our average velocity equation and we take our time values and shrink that gap smaller and smaller and smaller until the gap approaches zero. Then we arrive at our equation for instantaneous velocity. Practically, when we calculate instantaneous velocity in example questions, we’ll be taking derivatives of position as a function of time with respect to that variable time.

We’ve talked so far only about instantaneous velocity. But what about instantaneous speed? How are the two similar, and how are they different? It turns out that instantaneous speed is equal to the magnitude of instantaneous velocity. Recall that speed is a scalar while velocity is a vector. So while our instantaneous velocity could be negative, our instantaneous speed will always be positive or zero. Now that we have an idea for instantaneous velocity and instantaneous speed, let’s get some practice using these new concepts.

The position of an object changes as a function of time according to 𝑥 of 𝑡 equals negative three 𝑡 squared meters. What is the object’s velocity when 𝑡 equals one second? What is the object’s speed when 𝑡 equals one second?

Given a function describing an object’s position as a function of time, we want to know its velocity when 𝑡 is equal to one second and its speed at that same time. In other words, we want to solve for its instantaneous velocity and instantaneous speed. We can recall that an object’s instantaneous velocity is equal to the derivative of its position with respect to time. In our case, we’re given 𝑥 or position as a function of time and can plug in for that expression. When we take its derivative with respect to time, we find a result of negative six times 𝑡, now with units of meters per second. This is the instantaneous velocity of our object at a general time 𝑡. But we want to solve for that velocity when time is equal to one second. When 𝑡 is equal to one second, our instantaneous velocity is negative six times one meters per second, or negative six meters per second. That’s our instantaneous velocity when 𝑡 equals one second. Now what about its speed at that time?

We can recall that instantaneous speed is a scalar quantity. And it’s equal to the magnitude of instantaneous velocity. This means that the instantaneous speed of our object, when 𝑡 is equal to one second, is equal to the absolute value of negative six meters per second. This simplifies to six meters per second. That’s the object’s instantaneous speed when time equals one second.

That’s a bit about the difference between instantaneous speed and velocity. Now let’s look at an exercise that highlights the difference between average velocity and instantaneous velocity.

A particle’s position varies according to 𝑥 as a function of 𝑡 equals 3.0𝑡 squared plus 0.50𝑡 cubed meters. What is the particle’s instantaneous velocity when 𝑡 equals 2.0 seconds? What is the particle’s average velocity between 𝑡 equals 1.0 seconds and 𝑡 equals 3.0 seconds?

In part one of this exercise, we want to solve for instantaneous velocity when 𝑡 equals 2.0 seconds. Then in part two, we’ll solve for the average velocity of the particle over a time interval centered on that same time value. This one begins at 1.0 seconds and ends at 3.0 seconds. In calculating instantaneous velocity, we can recall that it’s equal to the time derivative of the position as a function of time of our particle of interest. So in our case, the instantaneous velocity of the particle as a function of time equals 𝑑 𝑑𝑡, the time derivative of position as a function of time which we’re given. When we plug in our equation for position as a function of time and take the time derivative. This derivative results in the expression 6.0𝑡 plus 1.50𝑡 squared in units of meters per second.

We’ve arrived at a general expression for instantaneous velocity. But we want to solve for the velocity at a particular time value, when 𝑡 equals 2.0 seconds. To calculate that value, we plug in a value of 2.0 seconds for 𝑡 in our expression. When we calculate this value, to two significant figures, our result is 18 meters per second. That’s the instantaneous velocity of our particle when 𝑡 equals 2.0 seconds. Next, we move on to solving for the average velocity of our particle over the time interval 1.0 to 3.0 seconds. This average velocity will be equal to the position of our particle at 3.0 seconds minus its position at 1.0 seconds divided by the time interval 3.0 minus 1.0 seconds. Or, in the denominator, 2.0 seconds.

To solve for the position of our particle at various times, we can use the expression given to us in the problem statement. When we plug in a value of 3.0 seconds for 𝑡 and calculate the position, we find a result of 40.5 meters. Which we insert for our position when 𝑡 equals 3.0 seconds. We then do the same thing for 𝑡 equals 1.0 seconds, plugging that value in to our expression and finding a result of 3.50 meters. Which we then insert for our position when 𝑡 equals 1.0 seconds. We’re now ready to calculate the average velocity of our particle over the time interval of interest. When we do and round the result to two significant figures, we find it’s equal to 19 meters per second. So our average velocity and our instantaneous velocity, even though they’re centered on the same time values, are not the same.

Let’s summarize what we’ve learned about instantaneous speed and instantaneous velocity. Instantaneous speed and instantaneous velocity are rates at a specific point in time. Instantaneous velocity is equal to the time rate of change of an object’s position, while instantaneous speed is equal to the magnitude of instantaneous velocity. And we can think of instantaneous velocity graphically by considering a position-versus-time curve. And consider that as our change in time shrinks smaller and smaller and smaller, we’re able to get a more finely resolved velocity. Until finally, as Δ𝑡 approaches zero in that limit, we have an instantaneous velocity at a particular time value. Instantaneous speed and instantaneous velocity are another tool we can use when analyzing object motion.

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