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### Angolo ottimale per una parte proiettile 3 - distanza orizzontale come funzione di angolo (e velocità)

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We now know how long the object is going to be in the
air, so we're ready to figure out how far
it's going to travel.
So we can just go back to kind of the core formula in all of
really kinematics, all of kind of projectile motion or
mechanical physical problems, and that's distance is equal
to rate times time.
Now, we're talking about the horizontal distance.
So our distance is going to be equal to-- what's our rate in
the horizontal direction?
We care about horizontal distance traveled, so our rate
needs to be the horizontal component of the velocity, or
the magnitude of the horizontal
component of the velocity.
And we figured that out in the first video.
That is s cosine of theta.
So let's write that down right here.
So our rate is s cosine of theta.
And how long will we be traveling at
this horizontal speed?
Well, we'll be going at that speed as long as
we are in the air.
So how long are we in the air?
Well, we figured that out in the last video.
We're going to be in the air this long-- 2 s sine of theta
divided by g.
So the time is going to be 2 s sine of theta over g.
So the total distance we're going to travel, pretty
straightforward, rate times time.
It's just the product of these two things.
And we could put all of the constants out front, so it's a
little bit clearer that it's a function of theta.
So we can write that the distance traveled-- let me do
that same green.
The distance traveled as a function of theta is equal
to-- I'll do that in this blue.
This s times 2s divided by g is-- I'll do it in a neutral
color actually.
This s times 2s divided by g is 2 times s squared over g.
So 2s squared over g times cosine of theta
times sine of theta.
So now we have a general function.
You give me an angle that I'm going to shoot something off
at and you give me the magnitude of its velocity, and
you give me the acceleration of gravity.
I guess if we were on some other planet, who knows?
And I will tell you exactly what the
horizontal distance is.