# Optimal angle for a projectile part 2: Hangtime | Physics | Khan Academy

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.

t = v/a. You can also use the formula: Vf = Vo + at and solve for t 🙂

why is the acceleration 10m/s squared?

wow i actually understand this XD

It's because acceleration is the same thing as the change in meters per second for each second – in other words, how much your speed in m/s changes per second.

You can express this as m/s divided by s, which gives you (m/s / s).

Since you divide m by s twice you get m/s² – meters divided by seconds divided by seconds again.

Can somebody please explain time me what does it mean that It's traveling so much and so much in a second SQUARED?

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Why will it take the same amount of time to fall the the ground?

Someone who didn't believe in gravity disliked this video xD