# Write an equation for the height as a function of time

So, little g over here, if you want to give it its direction, is negative. Let me get my calculator, bring it onto the screen.

Well, the first thing we can do is we could say, look, if the total time in the air was 5 seconds, that means that the time-- let me write it. Velocity The starting upward velocity will differ for each problem. Well, just from the definition of acceleration, change in velocity is equal to acceleration-- negative 9.

### Write an equation for the height as a function of time

Let me write the units down. This is displacement, and lets see You might be saying "Wait, clearly the force of gravity is dependent on the distance. Well, the first thing we can do is we could say, look, if the total time in the air was 5 seconds, that means that the time-- let me write it. We can express our final velocity in terms of our initial velocity and time. And it takes that same amount of time to now be accelerating in the downward direction back to that same magnitude of velocity but now in the downward direction. And what we do is one of us throws the ball, and the other one times how long the ball is in the air. Maybe we can get a little bit of a formula so maybe you can generalize it.

If you look at this previous video where we plotted displacement versus time, you see after 2 seconds the ball went from being on the ground-- or I guess the thrower's hand-- all the way to its peak height.

Remember that little g over there is all of these terms combined. Well, hopefully you found that entertaining. We can only calculate Vavg this way assuming constant acceleration. We will not examine spin, friction, or aerodynamics.

### H =- 16t 2 vt c

Our average velocity is your initial velocity plus your final velocity divided by 2, if we assume acceleration is constant. But if someone is able to throw the ball for 5 seconds in the air, they have thrown it 30 meters in the air. On Earth, gravity pulls on objects, fighting their ability to reach high heights and bringing them back down to the ground. 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. This is the graph for the previous example where we knew the initial velocity. Soccer balls get kicked, rocks and baseballs get tossed, and bullets and cannon balls all fly through the air with a characteristic upside-down u-shape. Now, what is our average velocity? Example 2 — A ball is thrown straight up from the top of a foot tall building with an initial speed of 48 feet per second. And then that gives you your initial velocity. So what is the distance-- or let me make it clear. Well, the first thing we can do is we could say, look, if the total time in the air was 5 seconds, that means that the time-- let me write it.

So this would be like a nine-story building. So let's play around with some numbers here just so we get a little bit more of a concrete sense.

The rock will have an initial velocity Vi of So it's literally just the arithmetic mean of your initial and final velocities.

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