Pointless physics question

UrbanSherpa · Post by **UrbanSherpa** » Wed Jan 21, 2009 12:25 pm

Someone asked a question the other day and I wondered if there were any engineers on this forum that could answer with the appropriate physics equation.

How high would an arrow (bolt) go if fired straight up in the air from a crossbow?
(Let's say an Exomax, for the most extreme example, and you can pick whatever bolt weight you want. Also, air friction must be accounted for, as we don't live in a vacuum)

Disclaimer: I am not in any way suggesting that you do this. The bolt would certainly fall from the sky and kill you every time! I'm just curious if any eggheads can figure it out.

Canabow · Post by **Canabow** » Wed Jan 21, 2009 12:50 pm

Not sure how high a crossbow would go but I know a reg bow will put it hi enough that you cant see it anymore. It never came straight down though moved about 30 to 50 yards everytime. We had set up a steel plate to hide under will we tried this.

Grizzly Adam · Post by **Grizzly Adam** » Wed Jan 21, 2009 12:53 pm

So ... what's your point?

Well, I can't say, that's for sure ... but wouldn't it depend a lot on variables like the humidity and temperature?

Pydpiper · Post by **Pydpiper** » Wed Jan 21, 2009 1:29 pm

My old fiberglass bow I had as a kid would go high enough to drift over to the next door neighbors house and stick in the vinyl of his hard top Cordoba. I remember it well.

sumner4991 · Post by **sumner4991** » Wed Jan 21, 2009 1:39 pm

Pydpiper wrote:My old fiberglass bow I had as a kid would go high enough to drift over to the next door neighbors house and stick in the vinyl of his hard top Cordoba. I remember it well.

Boo · Post by **Boo** » Wed Jan 21, 2009 1:41 pm

Pydpiper wrote:My old fiberglass bow I had as a kid would go high enough to drift over to the next door neighbors house and stick in the vinyl of his hard top Cordoba. I remember it well.

Why does this not surprise me???

Hey, did you guys hear? Pydpiper has high speed!

ecoaster · Post by **ecoaster** » Wed Jan 21, 2009 1:42 pm

I get 1902ft or 634yds, using 350fps.

Yes, I still have my Dynamic Motion text from university

. Assuming the arrow is a particle and there are no external forces acting on in such as friction.

Paid $100 for the damn thing and they would only buy it back for $10.

Pydpiper · Post by **Pydpiper** » Wed Jan 21, 2009 1:43 pm

Boo wrote:
Pydpiper wrote:My old fiberglass bow I had as a kid would go high enough to drift over to the next door neighbors house and stick in the vinyl of his hard top Cordoba. I remember it well.
Why does this not surprise me???
Hey, did you guys hear? Pydpiper has high speed!

I am thinking of changing my screen name to "ka-freaking-zoom"

catcher · Post by **catcher** » Wed Jan 21, 2009 2:01 pm

I told my Physics teacher that I will never use this stuff in real life. Now with this question this is twice in twenty years that it would have came in handy to pay attention. Man was I wrong.

ComfyBear · Post by **ComfyBear** » Wed Jan 21, 2009 2:39 pm

There's many factors and variables. that need to be quantified So I'm not going to give you the answer. However, to give you just an idea of what's involved, I'll outline a few things that one must take into account, when trying to arrive at the answer.

Sorry if it's long-winded.

First you need to know about GRAVITY DRAG

In astrodynamics and rocketry, gravity drag (or gravity losses) is the difference between the delta-v expended and the actual change in speed achieved by a spacecraft thrusting while in a gravitational field.
Gravity losses depend on the time over which thrust is applied as well the direction the thrust is applied in. Gravity losses as a proportion of delta-v are minimised if maximum thrust is applied for a short time, or if thrust is applied in a direction perpendicular to the local gravitational field. During the launch and ascent phase, however, thrust must be applied over a long period with a major component of thrust in the opposite direction to gravity, so gravity losses become significant. For example, to reach a speed of 7.8 km/s in low Earth orbit requires a delta-v of between 9 and 10 km/s. The additional 1.5 to 2 km/s delta-v is due to gravity losses and atmospheric drag.
Consider the simplified case of a vehicle with constant mass accelerating vertically upwards with a constant thrust per unit mass a in a gravitational field of strength g. The actual acceleration of the craft is a-g and it is using delta-v at a rate of a per unit time.

Over a time t the change in speed of the spacecraft is (a-g)t, whereas the delta-v expended is at. The gravity drag is the difference between these figures, which is gt. As a proportion of delta-v, the gravity drag is g/a.
A very large thrust over a very short time will achieve a desired speed increase with little gravity drag. On the other hand, if a is only slightly greater than g, the gravity drag is a large proportion of delta-v. Gravity drag can be described as the extra delta-v needed because of not being able to spend all the needed delta-v instantaneously.

However, since the launch trajectory does not remain vertical, VECTOR consideration has to be taken into account.

Thrust is a vector quantity, and the direction of the thrust has a large impact on the size of gravity losses. For instance, gravity drag on a rocket of mass m would reduce a 3mg thrust directed upward to an acceleration of 2g. However, the same 3mg thrust could be directed at such an angle that it had a 1mg upward component, completely canceled by gravity, and a horizontal component of 2.8mg, achieving a 2.8g horizontal acceleration.

As you can see, determining the final height can get complicated. Perhaps, by providing you with an example of a bullet, as the projectile, where the factors are known, and not accounting for air resistance (drag), which can vary according to atomspheric pressure. You can get a vague idea of what is possible.

The atmosphere has no precise boundary. It simply gets thinner as you go higher until it approaches zero density. A bullet fired straight upward would travel upward until v - gt=0 where t is the elapsed time in seconds, v is the muzzle velocity, and g = 9.8 meters/second^2. This does not take into account air resistance so it puts an upper maximum on how high the bullet would travel upward. To calculate the distance in meters d = 1/2 gt^2 where t^2 is t squared.

So, v=gt then d = 1/2 (v^2)/g

Using v = 340 m/s (a bullet with muzzle velocity = speed of sound)

We get d = 1/2 x 340^2 / 9.8 = 5897 meters (approx.)

So, a bullet with muzzle velocity equal to the speed of sound would travel upward about 6 kilometers. It would actually be a lot less due to air resistance. Even six kilometers is well under the level where the atmosphere has density close to zero.

A bullet with muzzle velocity = twice speed of sound would travel upward about 23.5 km and one with muzzle velocity = three times speed of sound would travel upward about 53 km. Those calculations do not take into account air resistance so the actual figures would be much less.

Hope that helps.

Grizzly Adam · Post by **Grizzly Adam** » Wed Jan 21, 2009 3:06 pm

UrbanSherpa wrote: How high would an arrow go if fired straight up in the air from a crossbow?

As a former student pilot, I must say that this is NOT one of the many hazards they warned us about in Ground School and practical training.

I've seen lots of sights from the window of a Cessna Skyhawk ... but I've never seen a crossbow arrow go flying by.

ComfyBear · Post by **ComfyBear** » Wed Jan 21, 2009 3:15 pm

Grizz, with a crossbow, you have nothing to worry about. A crossbow shooting an arrow at 350fps would only hit a maximum of 580.644 meters. In a vacuum, and alot less in real life.

350fps =106.68 m/s

d= .5 x 106.68 ^2/9.8 =580.644 meters = 1902.175 ft = 0.36079545 mile

UrbanSherpa · Post by **UrbanSherpa** » Wed Jan 21, 2009 3:35 pm

ComfyBear wrote:There's many factors and variables. that need to be quantified So I'm not going to give you the answer.......
Sorry if it's long-winded....... Hope that helps.

ComfyBear, This doesn't help at all.
The reason you're not going to give me an answer is because you don't know.
I, too, could have done a copy/paste of some rocket formulae from a British Army site.

I'd still like to know, though......anyone...........Bueller?

Pydpiper · Post by **Pydpiper** » Wed Jan 21, 2009 3:45 pm

Grizzly Adam wrote:
UrbanSherpa wrote: How high would an arrow go if fired straight up in the air from a crossbow?
As a former student pilot, I must say that this is NOT one of the many hazards they warned us about in Ground School and practical training.

I've seen lots of sights from the window of a Cessna Skyhawk ... but I've never seen a crossbow arrow go flying by.

That may have changed if they knew it was you in the drivers seat Adam..

"Hey, did that UFO say "cow whacker" on it?"

ComfyBear · Post by **ComfyBear** » Wed Jan 21, 2009 3:51 pm

How ungrateful !

The reason I don't is because unless you know and you can state all the variables that exist, there is NO definitive answer.

Furthermore, if the truth be know, to quote you, it's a pointless question; hardly worthy of an answer.

So if it's a question you must answer for your physics class....you're on your own.

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