If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

## 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.

## Want to join the conversation?

• what is the formula to find instantaneous velocity using calculus?
(30 votes)
• Using calculus, you would find the derivative of the function for the objects position to find the velocity at any given point.
(55 votes)
• How would you know at which angle to draw the tangent line on a graph to find the instantaneous velocity. By angle I mean how do you know how to draw that line if you only have one point? Thanks.
(31 votes)
• The point forces the angle. If you draw the tangent line any more or less steep it would no longer to a tangent. It would be a secant line.
(32 votes)
• I'm not sure I understand how to find instantaneous velocity and speed. I did not learn calculus yet and I don't know how to find the slope of a curve at a point in the curve. Of course when the average and instantaneous velocity are the same it's easy, but is there a way to figure it out that works for all scenarios?
(25 votes)
• I think if you did not learn calculus yet, then you will not be expected to calculate the instantaneous speed; more likely the speed over a given time. But it is a very interesting question... (and nicely presented)

I will chew it over further...
(22 votes)
• wahat do you mean by Vi?
(3 votes)
• initial velocity. if u find it confusing to use it, like in formulas such as Vf = Vi + at where it says final velocity is equal to initial velocity plus the product of acceleration and time, you simply use:-
v = u + at --> where 'v' is the final velocity, 'u' is initial velocity, 'a' is acceleration and 't' is time
(7 votes)
• Good video but it was so fast. Kinda wish Sal taught it
(12 votes)
• can you talk about the scalars i don't know much about it since you didn't really say how to figure out how to get the way the person went or how far you didn't talk about that just the vectors so in the next video can you please talk about scalars?
(3 votes)
• A scalar quantity is just the magnitude without a direction as far as i understand. When you give something a direction (like which way a force is acting or which way a car is going) then its a vector. Like on a car, you have a speedometer, not a velocimeter, because the car just tells you how fast you are going, not which direction.
(15 votes)
• at in video
i didnt understand why it's 0/0 ?
(4 votes)
• Is there an easy way to find the average speed of an object without using calculus? I haven't learned calculus yet.
(3 votes)
• Average speed = change in distance/change in time
(6 votes)
• At , what does the triangle with the "x" stand for?
(2 votes)
• It stands for 'change in displacement' or rather the displacement.
(6 votes)
• I couldn't understand the last case where acceleration was constant and the instantaneous velocity could be calculated by using the kinematical formula vf=vi+at. Can someone please explain clearly?
(4 votes)

## 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)