Main content

## Physics library

### Course: Physics library > Unit 1

Lesson 4: Kinematic formulas and projectile motion- Average velocity for constant acceleration
- Acceleration of aircraft carrier take-off
- Airbus A380 take-off distance
- Deriving displacement as a function of time, acceleration, and initial velocity
- Plotting projectile displacement, acceleration, and velocity
- Projectile height given time
- Deriving max projectile displacement given time
- Impact velocity from given height
- Viewing g as the value of Earth's gravitational field near the surface
- What are the kinematic formulas?
- Choosing kinematic equations
- Setting up problems with constant acceleration
- Kinematic formulas in one-dimension

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Deriving max projectile displacement given time

Deriving a formula for maximum projectile displacement as a function of elapsed time. Created by Sal Khan.

## Want to join the conversation?

- How do we know that the total time up is exactly half of the total time in the air?(28 votes)
- You can also prove it algebraically. Consider a initial speed V_o, at the highest point the velocity V = 0. Then using V = V_o + a*t(max height), we plug in a = -g and we see that 0 = V_o - gt or V_o/g = t(max height). Now lets consider total trip. We use Y = V_o*t + 1/2*a*t^2; Y = 0 here, because we are back down on the ground - > 0 = V_o*t - 1/2*g*t^2. So 0 = t(V_o -1/2*g*t). We get t = 0 or t = 2V_o/g. The first solution is at initial time it is on the ground. The second t is when it comes back down. We see that t(max height) = V_o/g and t = 2V_o/g so it is half the time. Hope this helps!(16 votes)

- Why isn't delta t squared divided by 4 as well? Sal just divides 4.9 by 4. The whole numerator should be divided by 4.(21 votes)
- I assume what you are refering to is at about time mark5:25. You can look at the s=(4.9 * v^2)/4 as s=1/4 * 4.9 * t^2 and both the 1/4 and the 4.9 are just numerical constants being multiplied with t^2 so you can combine them.

If you step back a step where he has s = ((4.9 * t)/2) * (t / 2) you can rewrite it as (1/2 * 4.9 * t) * (1/2 * t). Using the associtivge proerty of multiplication you can rewrite this as s = 1/2 * 1/2 * 4.9 * t * t. The 1/2 * 1/2 * 4.9 becomes the 1.225 and the t * t becomes t^2 givong you s = 1.225 * t^2.

I think you are confusing how to simplify (a + b) / c with (a * b) / c. (a + b) / c is equal to a/c + b/c where as (a * b) / c = a/c *b or a * b/c or a * b * 1/c.(39 votes)

- I was kinda puzzled when he changed 9.8 m/s^2 to 4.9 m/s^2. I know he just divided 9.8 by 2 but didn't understand why. So, while thinking about this problem I came up with another formula than the one Sal got at the end.

My question is, whether this formula is valid:

[ (9.8 * ▲T)/4 * ▲T/2 ](9 votes)- The entire equation he was working with was: DeltaV = -9.81 * DeltaT / 2. The first simplification he did was just to divide the -9.81 by the 2 so we then have DeltaV = -4.9 * DeltaT.

Your factor looks like it will simplify to (9.8 * DeltaT^2) / 8 which is 1.225 * DeltaT^2! Good Job!(8 votes)

- How would the ball's flightpath change with air resistance?(6 votes)
- If air resistance was included, the ball would experience drag, and the drag would produce a slower time. The greater the height or the more massive an object, the more the air resistance matters!

Hope that helps :)(10 votes)

- Why dosen't he use duration instead of Δt(4 votes)
- They're the exact same thing, and Δt is easier to use in equations, that and it's convention.(10 votes)

- Does this formula work if the object is thrown above the ground level and drops down to the ground level?(8 votes)
- No, it doesn't work in the case you mentioned, to the best of my understanding.(1 vote)

- Don't mass have any effect on the velocity or time?? If not, why?(3 votes)
- no, this is a common misconception. eventually it is the acceleration which matters, not the mass.

if a 1 ton brick, and a 1 kg block are dropped from the same height, neglecting air resistance, they would land at the same time.(8 votes)

- What if we projected a pistol bullet and cannon ball at same time, same height, same angle, also neglecting air resistance?

Will they both land on the ground at the same time?(5 votes)- yes! Projectile motion is independent of mass of the object involved! Moreover all objects which have the influence of acceleration due to gravity (g), will move in the same path! this is because 'g' does not depend on the mass of the object; g= GM/R^2, where G is gravitational constant, M is mass of earth and R is its Radius.(5 votes)

- This entire process is based on the assumption that you have a specific time. How could you solve if you didn't have the time?(4 votes)
- You could make a table for the different times but in the end you won't have a set answer.(5 votes)

- This is assuming the ball is launched directly from the ground. If I throw the ball from my hand, about a meter up, and we stop the stopwatch when the ball hits the ground, we can't assume it peaks at t/2. How would we go about it in this case?(4 votes)

## Video transcript

Just want to follow up
on the last video, where we threw balls in the air, and
saw how long they stayed up in the air. And we used that to figure
out how fast we initially threw the ball and how
high they went in the air. And in the last video, we
did it with specific numbers. In this video, I
just want to see if we can derive some
interesting formulas so that we can do the computations
really fast in our brains, while we're playing this game
out on some type of a field, and we don't necessarily
have any paper around. So let's say that the ball
is in the air for delta t. Delta t is equal
to time in the air. Then we know that
the time up is going to be half that, which is the
same thing as the time down. The time up is going to
be equal to delta t-- I'm going to do that
in the same color-- is going to be equal to the
time in the air divided by 2. So what was our
initial velocity? Well, all we have to
do is remind ourselves that the change
in velocity, which is the same thing as
the final velocity minus the initial velocity. So the final
velocity-- remember, we're just talking about half
of the path of this ball. So the time that
it gets released, and it's going at kind of
its maximum upward velocity, and it goes slower, and slower,
slower, all the way until it's stationary for just
a moment, and then it starts going down again. Now remember, the acceleration
is constant downwards this entire time. So what is the final
velocity if we just consider half of the time? Well, it's the time. It's 0. So it's going to be 0
minus our initial velocity, when it was taking off. That's our change in velocity. This is our change
in velocity, is going to be equal to the
acceleration of gravity, negative 9.8 meters
per second squared-- or the acceleration
due to gravity when an object is in free fall,
to be technically correct-- times the time that
we are going up. So times delta t up,
which is the same thing. I won't even write it. Delta t up is the same
thing as our total time in the air divided by 2. And so we get negative
the initial velocity is equal to-- this thing,
when you divide it by 2, is going to be 4.9 meters
per second squared-- we still have our negative
out front-- times our delta t. Remember, this is our
total time in the air, not just the time up. This is our total
time in the air. And then we multiply both
sides times a negative. We get our initial
velocity is just going to be equal to 4.9
meters per second squared times the total time that
we are in the air. Or you could say
it's going to be 9.8 meters per second squared
times half of the time that we're in the air. Either of those would get
you the same calculation. So let's figure out our total
distance, or the distance that we travel in the time up. So that'll give us
our peak distance. Remember that distance-- or
I should say displacement, in this situation--
displacement is equal to average velocity
times change in time. The change in time that we
care about is the time up. So that is our delta t over 2. Our total time divided by 2. This is our time up. And what's our average velocity? Well, the average
velocity, if we assume constant acceleration,
is your initial velocity plus your final velocity over 2. It's really just the
mean of the two things. Well, we know what our
initial velocity is. Our initial velocity is
this thing over here. So this is this thing over here. Our final velocity--
remember, we're just talking about
the first half of the time the
ball is in the air-- so it's final velocity is 0. We're talking when it gets
to this peak point, right over here-- that's from two
videos ago-- that peak point right over there. So our average
velocity is just going to be this stuff divided by 2. So it's going to be 4.9 meters
per second squared times delta t over 2. So this right here, this
is our average velocity. Velocity average. So let's stick that
back over here. So our maximum
displacement is going to be our average
velocity-- so that is 4.9 meters per second
squared-- times delta t, all of that over 2. And then we multiply it
again times the time up. So times delta t over 2 again. This is the same thing. These are the same thing. And then we can simplify it. Our maximum displacement is
equal to 4.9 meters per second squared times delta t
squared, all of that over 4. And then we can just
divide 4.9 divided by 4. 4.9 divided by 4 is-- let me
just get the calculator out. I don't want to do that
in my head, get this far and make a careless mistake. 4.9 divided by 4 is 1.225. So our maximum
displacement is going to be 1.225 times our total
time in the air squared, which is a pretty
straightforward calculation. So this is our max
displacement, kind of how high we get displaced. Right when that
ball is stationary, or has no net velocity, just
for a moment, and starts decelerating downwards. So we can use that. If a ball is in the air for
5 seconds-- we can verify our computation from
the last video-- our maximum displacement, 1.225,
times 5 squared, which is 25, will give us 30.625. That's what we got
in the last video. If the ball's in the
air for, I don't know, 2.3 seconds-- so
it's 1.225 times 2.3 squared-- then that means it
went 6.48 meters in the air. So anyway, I just wanted to give
you a simple expression that gives you the maximum
displacement from the ground, assuming air resistance
is negligible, as a function of the
total time in the air. I don't know. I find that pretty fun. And it's a neat game to play.