Instantaneous speed and velocity

pretend you were a physics student you were just getting out of class you were walking home when you remembered that there was a galaxy Wars Marathon on tonight so you do what every physics student would do run you're pretty motivated to get home so say you start running at 6 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 2 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 8 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 instantaneous velocity of this person at that particular moment in time note that this is different from the average velocity if your home was a thousand 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 at 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 is still just going to be 4 meters per second to the right or if you like positive 4 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 want to 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 want to 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 0/0 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 going to 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 going to 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 going to equal the instantaneous velocity at that point in time because the slope is going to 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 instant Dainius velocity V at any time T