Can a Bow Launch an Arrow at Supersonic Speeds?

[kroni]

Hello,

I am asking if from your point of view, is it possible to build a bow that can lunch arrow to a supersonic speed (speed superior to 350 m/s) ? I make some calculus, estimating cinetic energy needed but, is it possible to build a bow where this energy can be obtained ? In which material ?

PS : The bow have a classic size.

 

[A.T.]

I don't think anything in physics rules it out. It's more of an engineering problem, which might be unsolvable with current technology.

 

[rumborak]

A quick back-of-a-napkin calculation I just did seems to indicate that the archer would need to pull the equivalent of a 100kg weight, the weight of a full-sized adult. That's probably not feasible.
(assumptions: 20 grams arrow weight, 1m draw distance)

 

[sophiecentaur]

Perhaps a crossbow could get enough energy into your arrow. The arrow could be made extra light to make life a bit easier than rumborak's calculation suggests. Maybe ear protectors could be necessary to deal with the shock wave?

 

[boneh3ad]

Perhaps a crossbow could get enough energy into your arrow. The arrow could be made extra light to make life a bit easier than rumborak's calculation suggests. Maybe ear protectors could be necessary to deal with the shock wave?

There would be no shock wave or sonic boom observed by the archer. He/she is already behind the projectile.

 

[rumborak]

Perhaps a crossbow could get enough energy into your arrow. The arrow could be made extra light to make life a bit easier than rumborak's calculation suggests. Maybe ear protectors could be necessary to deal with the shock wave?

The arrow would have to be a special material too probably, as it would have to withstand the acceleration without snapping.

 

[Vanadium 50]

Like a lead ball? In a brass casing? Maybe propelled by gunpowder?

 

[JakeBrodskyPE]

If you could find the materials to do this safely, a crossbow could theoretically do the job. The problem is finding a string that can withstand the forces, a very low mass bolt (that's what a crossbow shoots) strong enough not to break from the acceleration, and the limbs capable of delivering the force without shattering. Believe me, if such a crossbow were possible, it would have been invented by now. I've seen high end crossbows capable of up to 130 meters/second, but not much more.

This is actually not a new problem. It is an exercise in material science more than anything else. As Vanadium 50 pointed out, this is why guns were invented.

 

[stedwards]

This is an interesting challenge. In the realm of nonexplosives, only a bullwhip seems to be doing this--no exotic materials required.

 

[rcgldr]

What about a (electric) rail gun that shoots arrows?

 

[sophiecentaur]

There would be no shock wave or sonic boom observed by the archer. He/she is already behind the projectile.

You could be right but the string would be at full speed as it passes within a few cm of your ear.

 

[JakeBrodskyPE]

You could be right but the string would be at full speed as it passes within a few cm of your ear.

Actually, the string wouldn't reach sonic speeds until it was well past your face (assuming you were endowed with the super-human strength to actually draw such a thing).

That said, even if this were a cross-bow, I wouldn't dream of firing something like this without ear protection.

And of course, the comic side of this is that as you reduce the air pressure, the speed of sound decreases. So to fire your supersonic arrow, first develop a bow that could shoot at 300 meters per second, and then take it to the summit of Mt. Everest...

 

[kroni]

Thanks for all your answer.

I am planning to build a crossbow. I hesitate between carbon fiber + epoxy resin and steel. I will make some computation to estimate the optimal form of the limbs.
I think that a good solution for the string is to use a piano wire.
I estimate that the piano wire will explode at the end of the propulsion step, when the arrow left it because limbs stop.
So it's a one shoot string but i can use a thinner one so the speed is higher.

What do you thing about this idea ?

 

[JakeBrodskyPE]

Thanks for all your answer.

I am planning to build a crossbow. I hesitate between carbon fiber + epoxy resin and steel. I will make some computation to estimate the optimal form of the limbs.
I think that a good solution for the string is to use a piano wire.
I estimate that the piano wire will explode at the end of the propulsion step, when the arrow left it because limbs stop.
So it's a one shoot string but i can use a thinner one so the speed is higher.

What do you thing about this idea ?

Even if this works, you should expect everything to self destruct, especially the limbs.

You might also want to spend time watching slow motion video of what happens to the limbs and string of a compound bow. It is a complex motion. Consider using extremely eccentric cams with the limbs, mounted on very heavy duty bearings. I would also use steel for the limbs because a carbon fiber system will probably shatter, where steel may not.

In another post

This is an interesting challenge. In the realm of nonexplosives, only a bullwhip seems to be doing this--no exotic materials required.

Allow me to add another non-explosive device that can break the sound barrier: Aircraft propellers. If the governor of an aircraft is not set properly, the engine may turn the propeller faster (typically 2800 RPM or more). This not only increases stress on the propeller significantly, it also doesn't yield any additional thrust. The reason is that the tips are breaking sonic speed. You won't notice much from inside the aircraft, but from the ground you'll hear a very loud, rasping roar as the plane of the propeller passes by.

Yet another possibility: Use pneumatics. See You could use the bow string on a very large piston that then launches the arrow...

 

[A.T.]

Allow me to add another non-explosive device that can break the sound barrier: Aircraft propellers. If the governor of an aircraft is not set properly...

Or the aircraft designer decides so...
http://en.wikipedia.org/wiki/Republic_XF-84H

 

[Nugatory]

I estimate that the piano wire will explode at the end of the propulsion step, when the arrow left it because limbs stop.

Be aware that the end of a snapped steel wire lashes around unpredictably and can do an amazing amount of damage... Slice an eyeball in two... snip off an earlobe... If you're going to experiment beyond what off-the-shelf technology offers, invest in a setup that allows to you to trigger your awesome toy infernal device superbow from a few meters away.

And treat the cocking process with at least as much respect (which is to say, a lot) as you would the compressed coil springs in a MacPherson strut.

 

[cjl]

And of course, the comic side of this is that as you reduce the air pressure, the speed of sound decreases. So to fire your supersonic arrow, first develop a bow that could shoot at 300 meters per second, and then take it to the summit of Mt. Everest...

This is a common misconception - the speed of sound in air is actually almost entirely independent of pressure. It basically only depends on the temperature of the air - speed of sound scales as the square root of the temperature. Yes, speed of sound decreases with altitude, but this is because the temperature decreases with altitude, not because the density or pressure does.

 

[kroni]

Clearly i am really concient of the dangerosity of such bow, wire and limbs. I will build a simple system to trigger the device without problem.

Thanks a lot for all your remark ! I will post picture if i have time to build it.

 

[JakeBrodskyPE]

This is a common misconception - the speed of sound in air is actually almost entirely independent of pressure. It basically only depends on the temperature of the air - speed of sound scales as the square root of the temperature. Yes, speed of sound decreases with altitude, but this is because the temperature decreases with altitude, not because the density or pressure does.

It is popular among ultrasonic ranging instrument vendors to use just temperature to estimate the speed of sound. As you say, temperature is the most sigificant effect. HOWEVER: The speed of sound is actually dependent upon air density. Density is a function of both pressure and temperature. In most applications, you tend not to see significant variations in barometric pressure, though the temperature can and does change significantly. In this hypothetical case, BOTH are significant.

By the way, although I am joking about the Everest quip, I did take the time to look up the speed of sound in a standard atmosphere at 30,000 feet.

 

[Bandersnatch]

The speed of sound is actually dependent upon air density.

Can you show us how? The only relevant equation I know is for the speed of sound in an ideal gas, and there's no density nor pressure factor in there since they conveniently cancel out:
http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe3.html
How much of a correction is needed for a non-ideal gas? Is it really at all significant?

 

[nasu]

It is popular among ultrasonic ranging instrument vendors to use just temperature to estimate the speed of sound. As you say, temperature is the most sigificant effect. HOWEVER: The speed of sound is actually dependent upon air density. Density is a function of both pressure and temperature. In most applications, you tend not to see significant variations in barometric pressure, though the temperature can and does change significantly. In this hypothetical case, BOTH are significant.

By the way, although I am joking about the Everest quip, I did take the time to look up the speed of sound in a standard atmosphere at 30,000 feet.

Yes, depends on density but also on the bulk modulus or some other measure of the gas' elasticity.
For ideal gas the bulk modulus is proportional with pressure. And so is density. The pressure cancels out from the speed of sound.

Here is an interesting graph showing the dependence on pressure for real air.
http://www.phy.mtu.edu/~suits/SpeedofSound.html
For dry air seems to be a slope of about 0.1m/s per atmosphere.
As the pressure changes in the atmospheric pressure are less than 1 atm, the ideal gas model is pretty good.
For humid air though the things look quite different.

 

[cjl]

It is popular among ultrasonic ranging instrument vendors to use just temperature to estimate the speed of sound. As you say, temperature is the most sigificant effect. HOWEVER: The speed of sound is actually dependent upon air density. Density is a function of both pressure and temperature. In most applications, you tend not to see significant variations in barometric pressure, though the temperature can and does change significantly. In this hypothetical case, BOTH are significant.

By the way, although I am joking about the Everest quip, I did take the time to look up the speed of sound in a standard atmosphere at 30,000 feet.

Sorry, but in the great majority of cases, density variation does not have a significant effect on sound speed in air. This includes the range from normal, sea-level conditions all the way up to the 80,000+ foot altitudes at which the SR-71 flew. There are special cases (dense gas dynamics, for example, at very high pressures) where the density can matter independent of temperature, but in the range where air behaves fairly close to an ideal gas, the speed of sound is density-independent.

This arises from the momentum equation - the complete derivation can be found here: http://www.sjsu.edu/faculty/watkins/sound.htm

 

[cjl]

Yes, depends on density but also on the bulk modulus or some other measure of the gas' elasticity.
For ideal gas the bulk modulus is proportional with pressure. And so is density. The pressure cancels out from the speed of sound.

Here is an interesting graph showing the dependence on pressure for real air.
http://www.phy.mtu.edu/~suits/SpeedofSound.html
For dry air seems to be a slope of about 0.1m/s per atmosphere.
As the pressure changes in the atmospheric pressure are less than 1 atm, the ideal gas model is pretty good.
For humid air though the things look quite different.

Pay attention to the scales though - for dry air, you're looking at variances of a couple tenths of a meter per second out of 340, so less than 0.1% error by assuming an ideal gas. Even for humid air though, it's only ~2m/s, which is still less than 1% error compared to the ideal gas model. Unless you need better than 1% accuracy for some reason, you really don't need to care about the sound speed variation with humidity or pressure for normal atmospheric conditions.

 

[nasu]

Oh, I meant that it goes the other way, when I said that it "looks quite different".
It is a small difference in both cases. I did not mean to imply that you need to care about it.:)

 

[JakeBrodskyPE]

Sorry, but in the great majority of cases, density variation does not have a significant effect on sound speed in air. This includes the range from normal, sea-level conditions all the way up to the 80,000+ foot altitudes at which the SR-71 flew. There are special cases (dense gas dynamics, for example, at very high pressures) where the density can matter independent of temperature, but in the range where air behaves fairly close to an ideal gas, the speed of sound is density-independent.

This arises from the momentum equation - the complete derivation can be found here: http://www.sjsu.edu/faculty/watkins/sound.htm

Apples and oranges. The derivation assumes that despite the pressure and temperature changes, the number of moles of gas remains constant. But that's not the case here.

Look up a table of standard atmosphere pressures and temperatures.

Look at the ratio of temperature to pressure at sea level and at 30,000 ft MSL (or about 9500 Meters MSL --I won't quibble much). They are different. This is because there isn't as much mass per unit volume at high altitude. The air at higher altitude IS less dense. How can it not be?

The point of this thought exercise was to change the environment so that the speed of sound would be slower. I think you must realize that at the top of Mount Everest, the atmosphere must be less dense than it is at sea level.

 

[nasu]

There was no argument about the change in air density with height.
It's just that the effect of density alone on the speed of sound is negligible.

If you don't like or understand the "theoretical" arguments, just look at the data.
If you measure the speed of sound at the same temperature, the effect of changes in pressure (or density) is minor. In this paper*, the speed of sound at about 0.5 atm was found to be 343.70 m/s and at 1 atm 343.43 m/s. The measurements were done at the same temperature. At 0.2 atm they found 343.80 m/s.

*R. L. ABBEY AND G. E. BARLOW, AUSTRALIAN J. SCI. RESEARCH A1, 175–189 (1948)
(reference and data quoted in "Handbook of the speed of sound in real gases", vol 3. by Allan Zuckerwar)

 

[cjl]

Apples and oranges. The derivation assumes that despite the pressure and temperature changes, the number of moles of gas remains constant. But that's not the case here.

Look up a table of standard atmosphere pressures and temperatures.

Look at the ratio of temperature to pressure at sea level and at 30,000 ft MSL (or about 9500 Meters MSL --I won't quibble much). They are different. This is because there isn't as much mass per unit volume at high altitude. The air at higher altitude IS less dense. How can it not be?

The point of this thought exercise was to change the environment so that the speed of sound would be slower. I think you must realize that at the top of Mount Everest, the atmosphere must be less dense than it is at sea level.

Of course the atmosphere is less dense at the top of Everest, but the effect of density on sound speed is negligible (and completely independent of the amount of gas present - sound speed is the same in a small enclosed container as it is in a large open space, so long as the conditions are the same).

Oh, and your last statement is a bit of a non-sequitur. If the point of the exercise is to change the environment so the speed of sound would be slower, drop the temperature. The fact that the atmosphere is less dense at the top of Everest is completely irrelevant to whether the sound speed is slower. I think you'll agree that the temperature tends to be much lower at the top of Everest than it typically is at sea level though, so the sound speed up there is definitely lower.

 

[Detective27]

Not possible... You would need a bow with spring constant 13,766 times the mass of the arrow considering there is no loss of kinetic energy

 

[Vanadium 50]

Not sensible. A spring constant is not measured in units of mass.

 

[kroni]

Hum ! I think the speed of sound in the air vary with density only at very low pressure. It's clear that empty space has no speed of sound, so there is a gap where the macro behaviour of matter disapear and micro effect like thermal effect are important.