Exploring the Relationship Between Mass and Velocity in Kinetic Energy

[Gara]

A 1 gram weight is moving at 1000 m/s

Would a 2 gram weight need to move at 250 m/s or 500 m/s to have the same kinetic energy? I'm guessing 250 m/s.

So would a 4 gram weight need to move at 125 m/s or 250 m/s to have the same kinetic energy?

And what object (out of the 1, 2, and 4 gram weights) requires the least amount of energy to move at it's suggested m/s, or is it the same?

I was reading about railguns (if you hadn't guessed by now) and I read somewhere (I never believe anything at face value) that a 0.5 gram weight was moved at some silly high speed (over 10 km/s) but because of it's weight, lacked momentum. Surely even a 0.5 gram needle type ammo moving at over 10 km/s would have no problem with momentum?

 

[AKG]

A 1 gram weight is moving at 1000 m/s

Would a 2 gram weight need to move at 250 m/s or 500 m/s to have the same kinetic energy? I'm guessing 250 m/s.

Why are you guessing? Do you know the equation for kinetic energy?

$$K = \frac{1}{2}mv^2$$

For your 1 gram mass, you get:

$$K = \frac{1}{2}(0.001)(1000)^2 = 500 J = 0.5 MJ$$

So, for a 2 gram mass to have the same kinetic energy:

$$500 J = \frac{1}{2}(0.002)v^2$$

$$v = 100\sqrt{50} m/s = 707.10678 m/s = 7 \times 10^2 \ m/s$$

You'll notice that I rounded to 1 significant digit as that's all that was given.

 

[Nenad]

I was reading about railguns (if you hadn't guessed by now) and I read somewhere (I never believe anything at face value) that a 0.5 gram weight was moved at some silly high speed (over 10 km/s) but because of it's weight, lacked momentum. Surely even a 0.5 gram needle type ammo moving at over 10 km/s would have no problem with momentum?

momentum is p = mv. If a gun fires a bullet at 10km/s and the bullet is 0.5g, then its momentum is (10km/s)(1000m/km)(0.0005kg) = 5kg*m/s. This is nothing. There is not enough momentum in this to pierce a piece of metal.

 

[Gara]

a .50 cal round

0.5 * (46 / 1 000) * (900^2) = 18 630

the 0.5 gram round

0.5 * (0.5 / 1 000) * (10 000^2) = 25 000

?

 

[Nenad]

a .50 cal round

0.5 * (46 / 1 000) * (900^2) = 18 630

the 0.5 gram round

0.5 * (0.5 / 1 000) * (10 000^2) = 25 000

?

what are you doing? What units are you working in?

 

[Gara]

1/2 * (0.5 gram / 1000 [to make it kilograms]) * 10,000 [meters per second]^2) = 25,000 jouels, or something.

 

[Nenad]

yes, that is kinetic energy: $$E = 1/2mv^2$$
but momentum is NOT kinetic energy, it is the product of the mass with the velocity:
$$p = mv$$

 

[Gara]

I see. So while it would have a lot of kinetic energy, it would lack momentum? Right.

 

[Nenad]

yes, this could happen, but the projectile lacks both, because of its minute mass. It only has: $$E_k=0.5MJ \\\ p=5kg*m/s$$ This would have no serious effect on a person. It woulden't harm a dog.

 

[kirkmcloren]

yes, this could happen, but the projectile lacks both, because of its minute mass. It only has: $$E_k=0.5MJ \\\ p=5kg*m/s$$ This would have no serious effect on a person. It woulden't harm a dog.

Once the mach front exceeds the strength of the material a hell of a lot of damage occurs. That is why projectiles in excess of 3000 feet per second are so dangerous. a 55 grain 223 bullet can destroy your arm yet a .308 with triple the mass may leave less damage. At 10km/s a phonograph needle would liquify meat leaving a huge wound channel.

 

[pervect]

I see. So while it would have a lot of kinetic energy, it would lack momentum? Right.

Right, but it would still NOT be good to be hit by one of these things. If you go to http://www.onlineconversion.com/energy.htm
you can convert this amount of energy into high explosive (tons of TNT)

25,000 joules turns out to be about 6gm of tnt.

Low-mass high velocity impacts like the hypothetical railgun, or like a meteor impact,
tend to act like an explosion, so thinking of the impact as being like having this amount of high explosive detonate on the person in a surface explosion is a reasonable model of what happens when you get hit.

Offhand, I don't know the exact effect of exploding 6gm of TNT on a person or animal, but I don't think it would be particularly good. Certainly it would not be something that would be "safe to do to a dog".

 

[Gara]

"Low-mass high velocity impacts like the hypothetical railgun"

the railgun is real.

 

[pervect]

the railgun is real.

Is the specific railgun you were talking about with a .5 gm slug and a 10km/sec velocity real, or were those particular figures hypothetical?

I'll agree that railguns are "real", though AFAIK they are still experimental one shot laboratory devices as opposed to deployed systems. I may well be "behind the times".

I'd expect even an experimental railgun to be designed for anti-tank use, and that it would have a heavier projectile, lower velocity, and higher projectile mass than the specific figures you quoted.

For instance google finds mention of a late 80's DARPA 9 Megajoule railgun, with a 2.5-4 km/sec velocity, which implies a 1-3 kg projectile.

http://www.fas.org/man/dod-101/sys/land/docs/5fcs97.pdf

One problem I'd expect the .5gm slug 10km/sec design to have is large losses in kinetic energy due to air friction, which goes up as the square of the velocity (power lost due to friction goes up even faster, as the cube). My guesstimate is that this would imply a fairly short effective range for the weapon, though I haven't attempted to actually calculate or guess at any of the exact figures.

 

[Gara]

Yes, the gun as well as the ammo's weight and speed are real, and yes, it proberly has almost no use at such low weight. :)

 

[Nenad]

but see, it does have use. If the rail gun's ammo is shaped properly, and very small, it can be danger to humans. The impact would put a lot of pressure on the skin where it hits and it would pierce it.

 

[pervect]

Yes, the gun as well as the ammo's weight and speed are real, and yes, it proberly has almost no use at such low weight. :)

Any references or URL's? I'd be interested in reading more about it.

 

[pervect]

but see, it does have use. If the rail gun's ammo is shaped properly, and very small, it can be danger to humans. The impact would put a lot of pressure on the skin where it hits and it would pierce it.

One of the biggest question is whether the ammo (penetrator) is going to fracture when it hits its target. If you attempt to make the round too long and skinny, you can pretty much guarantee fracture. At a high enough velocity, you won't be able to avoid fracture in any event. I initially thought that 10 km/sec would be too high to maintain rod integrity, but it looks like I was probably wrong.

I did some googling and found some interesting results for "water filled targets" and tungsten rod penetrators which would be a good analog for say, a human body

http://sherpa.sandia.gov/9231home/pdfpapers/ballistics98_longrod.pdf

The highest velocity reached was only about 5km/sec. The rods were a lot longer and heavier, but for much of the aerodynamics only the scale factor should matter. The rods did tend to erode quickly, and all the rods broke when the path thorugh the water was long enough, though the exact mechanism wasn't known. I don't think the .5 gm round would survive intact a passage through a human body. (It would erode or break when you scale the experimental results above). Though the good news is that such a round would probably deliver most of its considerable energy to the target.

In the actual experiment, with a 3:1 length ratio the rods survived, with a 15:1 length ratio the rods fractured, the computer code wasn't predicting the fracture properly.

There's also an interesting aerodynamic effect - a sharply pointed body is going to have low aerodynamic friction (which is good), but the tip is going to tend to melt (bad). The "stagnation point heating" turns out to be inversely proportional to the sqaure root of the radius of curvature of the tip.
(I don't recall where I first read this, but google finds the URL below as confirmation) So a very sharp tip (small radius) is probably not a good idea. Especially since the "sharpness" won't last long once the penetrator hits at those sorts of velocities.

http://www.ucsusa.org/documents/CM_apF-J.pdf

So at this point my best guess at a projectile design would be a tungsten (or depleted uranium) - because one needs a hard and dense penetrator 3x as long as it is wide, with a hemispherically curved tip, and some sort of stabilization at the rear - probably fins - to keep it from tumbling.

 

[Gara]

http://www.powerlabs.org/railgun.htm

"16,000m/s firing of a 0.1 gram object"

"and they can also propel objects of very sizeable masses to equally impressive velocities, such as in the picture to the left, (http://www.powerlabs.org/images/mrailgun.jpg ) where Maxwell Laboratories' 32Megajoule gun fires a 1.6kilogram projectile at 3300m/s (that's 9megajoules of kinetic energy!)"

Whoa.

As for the railgun on that site, he uses "...a sharpened tungsten spear placed inside a tapered Teflon projectile. Two layers of aluminum foil were crumpled behind it..."

http://www.powerlabs.org/images/plasmaprojectil1aftr.jpg
http://www.powerlabs.org/images/plasmaprojectil2.jpg

 

[MoonUnit]

Another interesting railgun document where they talk about fitting ships with railguns capable of firing about 6 rounds per minute I think, over a range of 250 miles, with a LOT of destructive potential (although these are pretty big projectiles):

http://www.battelle.org/navy/railguns.pdf