Formula wanted for how far you can jump off a building?

Factual Questions
1

A friend of mine recently moved into a high-rise apartment building with a swimming pool. While on the roof admiring the view, he asked if I thought a person could land in the swimming pool if they got a running jump off the roof (it is about 25 feet from the building to the pool). I didn’t think that the extra height (about 20 floors) would provide the addition carry needed to make the jump. Curious to find out the answer and not willing to have my friend attempt the jump; I decide to ask here. (Add I figured the bottle of tequila to convince him to jump would make it hard for him to walk much less run and jump).

So what is the formula to figure the horizontal distance traveled before the body hits the ground given the height of a platform and the speed and weight of the body?

2

Well if you don’t mind hitting the water at 80 mph you can easily traverse the 25 feet out to the pool.

The formulae are:

Vertical distance traveled under gravity = 16 time[sup]2[/sup]
(This works out to about 3.5 seconds for the fall time, give or take.)

To travel 25 feet in 3.5 seconds you have to be going around 5 mph, say a good fast trot.

Watch out for that last few feet though. Water am hard at that speed…

3

d=v*sqrt(2h/g)
d= distance traveled
v= horizontal velocity
h= height of platform
g= acceleration due to gravity
weight is not a significant factor for a human being (although obviously it would be for something like a feather)

If d, v, and h are given in terms of meters and seconds, g~=9.8. If each floor is 2.5 m, then d~=3v. So to go 8m (about 25 feet), you’d have to be traveling at 2.67 m/s, or about 5.6 mph. Definitely doable.

4

Sometime before I moved into this building, which is two storeys, a guy tried to jump from the second floor walkway and into the pool. He died. See, it wasn’t the height, but his aim. He hit his head on the side of the pool. Alcohol was involved.

5

Since this is about one quarter of the speed that an Olympic long jumper is going at take off I’d paint a cross about 3 feet out from the building for your friend to aim at.

None of these calculations allow for the fact that once you leave the roof you decelarate real fast. You finish the trip travelling vertically. If this was not true a 10 second sprinter could jump off, and using the same calculation, travel 100 feet horizontally.

6

I meant to add - he can experiment on a diving tower. He’ll find he can jump out about as far as he can on dry land and that won’t be 25 feet.

7

ignoring air resistance, this isn’t true. there is no force in the x-direction, so he stays going at a constant velocity.

8

And can we ignore air-resistance?

Perhaps we should ignore gravity – makes the maths a whole lot simpler.

9

It’s much easier if you use a [url"http://half-life.gamehelp.com/opfor/weapons/weapon_gauss.html"]guass gun to shoot yourself off the bulding.

How you do this is pretty simple. You face with your back to the swimming pool. Charge the gun up for 1 second per foot you need to be shot backwards, and let go of the trigger. Try to aim at a place in the floor that a tenant will not be likely to be standing as it will travel through several floors.

If you aren’t worried about the building itself nor the tenants, this feat can be replicated using a 1920’s style death ray.

10

Err, try this: Guass Gun

Preview is your friend.

11

To test the above (~5mph) calculations, just go up to the roof and run towards the pool edge with a lemon. Let it go just before you plunge to your death, and see where it lands.
If you can come up with a good way to launch the lemon at the proper speed horizontally, that would work better.

12

Won’t somebody think of the fruit?
Actually Floyd, that’s a good idea. Welcome to the boards, and all that.

13

What am I saying, you’ve been here for years and posted TWICE!

You’re not a complete whacko with a 1920s style death ray are you?

14

Unless you encounter significant air resistance at 5 mph (which I don’t), then yes, we can pretty safely ignore air resistance.

Granted, there’s going to be some effect even on the horizontal speed once you’ve reached an appreciable downward speed, but I’m pretty sure the difference is less than a meter.

Considering a 50m building, if we ignored air resistance, your downward speed is about 30 m/s when you hit. Terminal velocity of a human is usually more like 50 m/s, so we’re not wildly off in our time calculation.

I’d guess that you’re more likely to go farther using air resistance - the slowdown in your horizontal speed is probably less than the effect of more time falling (due to decreasing downward acceleration).

Strange that a Gauss gun would have such a kick in a game - most of the time ‘Gauss gun’ refers to a coil gun that doesn’t have a strong recoil.

15

Actually, turns out air resistance doesn’t make a whole lot of difference. I ginned up a time-step spreadsheet (remembering to use the vector velocity in the drag calculation); here’s the results:


                without AR     with AR
Fall dist        200ft         200ft
Start Horiz Vel  7ft/sec       7ft/sec
Time             3.5sec        3.57sec
End Horiz dist   24.5ft        23.95ft
End Vert Vel     112ft/sec     106ft/sec
End Vert Acc     32ft/s/s      25.1ft/s/s

(for ref: terminal velocity calculated ~=220ft/sec) The 200 ft is just not enough to allow air resistance to have a big effect.

16

He’s spent the last three years running up and then posting lemons at the last minute. Just to see what happens.

Can’t be too careful.

17

As you can see from zut’s post, you finish the trip travelling at ~7fps horizontally and ~100fps vertically. This gives you an angle of 86 degrees. Almost, but not quite vertical, and not because your horizontal speed has slowed, but because you’re going so fast vertically.

18

zut, are you sure that the final distance with air resistance isn’t 24.95, not 23.95? I wouldn’t be surprised if I’d guessed wrong, but it seems strange that without a lot of horizontal speed you end up almost a whole foot off the value based on time and initial speed.

I should also correct myself - terminal velocity is more like 60 m/s than 50 m/s.

19

The building is 20 stories which is 240 feet. My husband is a swimmer. There is a formula for pool depth to rise,for diving, but it only goes to 30 feet rise. At 30 feet the pool needs to 45 feet deep to avoid injury hitting the bottom.
Also, choppy water is “softer” than flat water. A pool would be flat as in concrete at that height.
I’ve cared for water jumpers when I worked trauma. (Off the Coronado Bridge in San Diego) Not pretty. I don’t recommend putting your number games to the test with a human!

20

Well, it’s only half-a-foot (23.95 versus 24.5), not so big percentage-wise.

Sorry, I posted in a hurry and wasn’t too clear. The numbers I ginned up above are using 7fps as a starting velocity.

Here, lemmee do this again with another sig fig, a 5mph starting velocity, and slightly higher air resistance to match panamajack’s terminal velocity:

Results with 5mph (7.334fps) starting velocity; vertical terminal velocity = 197fps


                Without AR    With AR
Fall Dist        200ft        200ft
Start Horiz Vel  7.334fps     7.334fps
Time to Fall     3.500sec     3.597sec
End Horiz Dist   25.664ft     24.911ft
End Horiz Vel    7.334fps     6.211fps
End Vert Vel     112.7fps     104.1fps
End Vert Acc     32.2f/s/s    23.3f/s/s

So, yeah, over half-a-foot of horizontal travel difference, but note that the horizontal velocity has dropped a full fps, too. A chart:


Time    Horiz Vel
0.0     7.334fps
0.5     7.300fps
1.0     7.226fps
1.5     7.107fps
2.0     6.947fps
2.5     6.749fps
3.0     6.519fps
3.5     6.263fps

A shorter building would have less distance differential: for example, with a 100ft building, you get 17.815ft with air resistance versus 18.093ft without.

And, just for comparison, jumping off the Empire State Building:


                Without AR    With AR
Fall Dist        1250ft       1250ft
Start Horiz Vel  7.334fps     7.334fps
Time to Fall     8.786sec     10.364sec
End Horiz Dist   64.435ft     53.845ft
End Horiz Vel    7.334fps     2.579fps
End Vert Vel     282.9fps     183.7fps
End Vert Acc     32.2f/s/s    4.01f/s/s

Note that all these numbers are generated using a time-step spreadsheet, so the usual computational inaccuracies apply (for example, the end horizontal distance is consistantly slightly underpredicted); the sig figs listed imply an accuracy that is not present. The final number ought to be reasonably accurate, though; additionally, comparing two runs (with versus without air resistance, for example) should be pretty valid, as inaccuracies are consistent from one run tothe next.