Optimal angle for a projectile
Optimal angle for a projectile part 2 - Hangtime Optimal angle for a projectile part 2 - Hangtime
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- Let's figure out how long this object is going to be in the
- air given that its vertical velocity, or the magnitude of
- the vertical velocity is s sine of theta.
- So its speed in the vertical direction is s sine of theta.
- So how long is it going to be in the air?
- Well if I told you that something is going upwards at
- 10 meters per second and gravity is decelerating it at
- 10 meters per second squared.
- So every second it's going to slow it down by 10 meters per
- second, how long will it take for that object to get to 0,
- to stop moving?
- Let me write that down.
- Let's say that some object is moving upwards at 10 meters
- per second.
- And let's say that the gravity is slowing it down.
- Slowing it down at 10 meters per second.
- So every second that goes by, it'll slow this thing down by
- 10 meters per second.
- Well, it'll take it exactly 1 second to make it go from 10
- meters per second to 0 meters per second.
- And then it's going to be at some height in the air, and
- then the thing's going to start accelerating.
- Gravity is going to start accelerating it downward.
- And then it'll take another second for it to go from 0--
- from having no velocity, to having 10
- meters per second again.
- So in this case, it'll take the time in the air-- we could
- say time sub air, I guess, is going to be equal to this 10
- meters per second, your velocity.
- 10 meters per second divided by the acceleration.
- Divided by this 10 meters per second.
- 10 meters per second times 2.
- This is how long it'll take for the object to go from 10
- meters per second to 0 at some point in the air.
- And then it's going to take the exact same amount of time
- for it to fall back to the ground.
- So times 2.
- If the object was moving upwards at 20 meters per
- second and gravity is still slowing it down at 10 meters
- per second per second, then it's going to take 2 seconds.
- If this was 20, then this would be 20.
- And it'll take 2 seconds to slow it down to 0 and then 2
- more seconds until it hits the ground again.
- For it to speed back up as it approaches the ground.
- So no matter what your upward velocity, the time in the air
- is going to be your speed, your vertical speed, divided
- by the acceleration of gravity.
- And this is the amount of time it's going to take you to go
- from this point to that point.
- To have some vertical velocity and then slow down to 0.
- And it's going to take the exact same amount of time for
- you to speed back up by gravity and get to your
- original speed.
- We're assuming no air resistance.
- So it's kind of a pure problem.
- So this is the time up, the time down is going to be the
- same thing.
- So we can multiply that by 2.
- Now, we already know what the vertical component for our
- problem is.
- It is s sine of theta.
- So we could just substitute that back in there.
- And we know how long we're going to be in the air.
- The time in the air is going to be our speed-- or I should
- maybe put the 2 out front.
- 2 times s sine theta.
- Let me make it clear.
- This 2 right here is this 2 right there.
- All of that over the acceleration of gravity.
- So if you told me that I'm shooting this object off at--
- I don't know-- 100 meters per second.
- So if this is 100 meters per second and if theta were-- I
- don't know-- let's say theta were 30 degrees, then sine of
- theta would be 1/2.
- So it'd be 100 meters per second times 1/2 divided by
- the acceleration of gravity times 2 would tell you exactly
- how long you would be in the air.
- How long it takes to go up all the way, become stationery,
- and then fall back down to the ground.
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