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## AP®︎/College Physics 1

### Course: AP®︎/College Physics 1>Unit 1

Lesson 4: Velocity and speed from graphs

# Instantaneous speed and velocity

Instantaneous speed is a measurement of how fast an object is moving at that particular moment. Instantaneous velocity is a vector quantity that includes both the speed and the direction in which the object is moving. Learn how to find an object’s instantaneous speed or velocity in three ways - by using calculus, by looking at the slope of a given point on a graph of an object’s rate vs. time, or by using kinematic formulas if the object’s acceleration is constant. Created by David SantoPietro.

## Video transcript

- [Instructor] Pretend you are a physics student. You are just getting out of class. You were walking home when you remembered that there was a Galaxy Wars marathon on tonight, so you'd do what every physics student would do: run. You're pretty motivated to get home, so say you start running at six meters per second. Maybe it's been a while since the last time you ran, so you have to slow down a little bit to two meters per second. When you get a little closer to home, you say: "No, Captain Antares wouldn't give up "and I'm not giving up either", and you start running at eight meters per second and you make it home just in time for the opening music. These numbers are values of the instantaneous speed. The instantaneous speed is the speed of an object at a particular moment in time. And if you include the direction with that speed, you get the instantaneous velocity. In other words, eight meters per second to the right was the instantaneously velocity of this person at that particular moment in time. Note that this is different from the average velocity. If your home was 1,000 meters away from school and it took you a total of 200 seconds to get there, your average velocity would be five meters per second, which doesn't necessarily equal the instantaneous velocities at particular points on your trip. In other words, let's say you jogged 60 meters in a time of 15 seconds. During this time you were speeding up and slowing down and changing your speed at every moment. Regardless of the speeding up or slowing down that took place during this path, your average velocity's still just gonna be four meters per second to the right; or, if you like, positive four meters per second. Say you wanted to know the instantaneous velocity at a particular point in time during this trip. In that case, you'd wanna find a smaller displacement over a shorter time interval that's centered at that point where you're trying to find the instantaneous velocity. This would give you a better value for the instantaneous velocity but it still wouldn't be perfect. In order to better zero-in on the instantaneous velocity, we could choose an even smaller displacement over that even shorter time interval. But we're gonna run into a problem here because if you wanna find a perfect value for the instantaneous velocity, you'd have to take an infinitesimally-small displacement divided by an infinitesimally-small time interval. But that's basically zero divided by zero, and for a long time no one could make any sense of this. In fact, since defining motion at a particular point in time seemed impossible, it made some ancient Greeks question whether motion had any meaning at all. They wondered whether motion was just an illusion. Eventually, Sir Isaac Newton developed a whole new way to do math that lets you figure out answers to these types of questions. Today we call the math that Newton invented calculus. So if you were to ask a physicist: "What's the formula for the instantaneous velocity?", he or she would probably give you a formula that involves calculus. But, in case some of you haven't taken calculus yet, I'm gonna show you a few ways to find the instantaneous velocity that don't require the use of calculus. The first way is so simple that it's kind of obvious. If you're lucky enough to have a case where the velocity of an object doesn't change, then the formula for average velocity is just gonna give you the same number as the instantaneous velocity at any point in time. If your velocity is changing, one way you can find the instantaneous velocity is by looking at the motion on an x-versus-t graph. The slope at any particular point on this position-versus-time graph is gonna equal the instantaneous velocity at that point in time because the slope is gonna give the instantaneous rate at which x is changing with respect to time. A third way to find the instantaneous velocity is for another special case where the acceleration is constant. If the acceleration is constant, you can use the Kinematic Formulas to find the instantaneous velocity, v, at any time, t. (electronic music)