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So it’s going to be plus, one half times v f, times our final velocity. Maybe if you were under time pressure you would want to be able to whip this out, but the important thing, so you remember how to do this when you are 30 or 40 or 50 or when you are an engineer and you are trying to send a rocket into space and you don’t have a physics book to look it up, is that it comes from the simple displacement is equal to average velocity times change in time and we assume constant acceleration, and you can just derive the rest of this.

Once again when were are dealing with objects not too far from the center of the earth we can make that assumption. So I’m going to do it both, using the variables, and using the concrete numbers.

### Gennemsnitlig hastighed for konstant acceleration (video) | Khan Academy

Or I have a constant slope here. The distance we traveled is equal to one half of the initial velocity plus the final velocity. And so I’m going to define this as something new.

This right here is the average velocity. So if I want to see how much faster have I gone, I multiply the number of seconds– I’ll just multiply the number of seconds that go by, times my acceleration. Our initial velocity was five meters per second.

And this tells us the distance. And let’s see how we can simplify this Well these two terms remember we are just dealing with the average velocity here These two terms if you combine them become 2 times initial velocity 2vi. So one, two, three, four. I will get to my final velocity. After two seconds we will be at nine meters per second.

Easy to figure out the area of a triangle. So what is v i minus one half times v i. But the only reason why I can just take the starting velocity and the ending velocity and, adding them together, and then divide by two.

So we have five meters per second, plus four times two is eight. The whole reason why I did this is when we look at the g it really comes from the universal law of gravitation. Udgavd you multiply how much time passed times acceleration.

Because it’s tdien meters per second, per second.

You know that it is specifying what the magnitude and the direction is. And I want to take a pause here.

The other thing I want to make clear: Video udskrift What I want to do with this video is think about what happens to some type of projectile, maybe a ball or rock, if I were to throw it straight up into the air.

Well I’m starting at five meters per second. So the total distance we traveled is 20 plus 16, is 36 meters. So we can rewrite this expression as the initial velocity plus something over 2.

## Deriving displacement as a function of time, acceleration, and initial velocity

It’s the arithmetic mean of these two numbers. But it’s not always the assumption. After three seconds we will be 11 meters per second. This is the area of this. We have to take it by half because a triangle is only half of that rectangle.

Or another way to think about it, is the slope of this velocity line is my constant acceleration. We are going to leave it right over here. You can break it down into this blue part, this rectangle right over here. So you have 13 plus 5 is equal to So my final velocity is going to be right over there.

In the next video we are going to use this formula we just derived. Which is times our final velocity minus our initial velocity.