Okay, let's talk about velocity We just saw what displacement looks like let's talk about velocity and specifically let's talk about average velocity. So, average velocity– it's still vector but for simplicity we just write it with a bar on top without that arrow and average velocity is simply this: Delta r over delta t, okay. Delta R is a vector delta T is of course the scalar. V is therefore a vector. Now that's your average velocity, and we've done a bunch of problems calculating average velocity but let's review. Let's say you drive from San Diego to LA and then you drive back to San Diego. What's your average velocity for that whole trip? I'm gonna ask you this question: What's your average velocity for the whole trip? Yeah. >> It's your final velocity when you get to Los Angeles minus your initial velocity when you started in San Diego. >> Okay >> Divided by time. >> Okay all right I think maybe we should go back to this definition for average velocity and see what this definition says. This definition would be the following: r f minus r i divided by delta t. What is your r f in this case? >> Your final position. >> Your final position, which is? >> Los Angeles? >> No, if we went from San Diego to LA and back to San Diego– >> San Diego. >> What's our final position? It's San Diego. What's our initial position? >> (student speaking) >> San Diego. So what is your average velocity for the whole trip? >> Zero. >> Zero, which seems a little weird, right? How could your average velocity be zero for the whole trip? It's because it's a vector. And so what's important is the displacement for your whole trip. r f minus r i. If you end at the same point you started from, your average velocity is always zero. And it kind of makes sense, right? I drive 60 miles per hour north up to LA but then I drive minus 60 miles per hour coming the other way. And so it all averages out to zero. Now, different question of course is what's your average speed for the whole trip? And based on the example the example that I just gave you, that would be 60 miles per hour. That's a little different, but velocity is a vector and so you have to worry about the end points compared to the starting points for that, okay. What about instantaneous velocity? Well that one we can write like this. Also a vector it's just not an average so we don't put the bar on it. And instantaneous velocity is when you're driving along and you look at your speedometer in your car, that is your instantaneous velocity and so it is the limit of delta r over delta t as delta T goes to zero. Namely I'm gonna take this measurement over a very short time. Anybody in calculus here? Yeah so what is this quantity that we're talking about right here? >> It's the average speed. Okay, but it has a very special name in Calculus, which is what? I'll give you a hint: it's not the integral it's the... >> Derivative. >> Derivative, yeah this is the derivative. As you take the limit of this function as delta T goes to zero, this just becomes a derivative. Okay, dr/dt Alright, maybe we should try an example of that. Let's say that we have the following: Let's give it r is equal to 1/2 a t squared and we are moving in one dimension i hat. What is v? Well we just have to take a derivative. So v is going to be dr/dt which is d/dt the derivative of 1/2 a t squared i hat What is the derivative of 1/2 a t squared? Somebody raise your hand. Yeah. >> a t >> a t, right. I pull down the exponent that 2 cancels with the 1/2 I go a t to the 2 minus 1 which is just a t and so I just get that. Are we done? Is that right or do I have to add something else there? Yeah. >> You have to add in the direction, which is i hat. >> Yeah, exactly. We got to hang on to that i hat. The velocity is a t i hat, if you have a vector sign on the left side of the equation, you have to have a vector sign somewhere on the right side of the equation. Ok good. Now let's talk about two-dimensional motion and let's take a look at how we can do an example like this in two dimensions.