Thermodynamics of a cannon ball / musket ball shot bullet

[darkdave3000]

I am a computer scientist currently writing a 3D simulation for high speed projectiles both sub and supersonic for various shapes and sizes of projectiles.

For now I am focused on completing simulation of a simple model of a round ball shaped projectile (ie a cannonball or musket shot).

I like my program to visually simulate the flow of heat due to air friction with the projectile but need direction to know what wikipedia source to read. I want to simulate friction force being converted into heat and simulate this transfer of infra red radiation from interaction of ball and air to air and to ball and from ball back to air.

Hope that makes sense. I am confident I can find the relevant formula in publicly available articles on thermodynamics and college level physics as it sounds like a straight forward problem to me.

How do I calculate the percentage of heat that is absorbed by the ball vs absorbed by the air when generated by the interaction of the two substances?

Assuming that the ball is made of well known metallic materials such as alluminium where there is plenty of supporting properties and experimental data of the material to help with the above question.

If supersonic speeds will make this too hard please then just assume subsonic speeds, perhaps of a typical 6lb cannon from the 1800's.

Assume also that Drag coefficient is 0.47

The solution does not need to be 100% accurate, a simplified solution for 90-95% accuracy will surfice.

 

[Andrew Mason]

Essentially, the loss of kinetic energy of the ball as a function of distance or time will depend on how that loss of energy is distributed between the air molecules and the ball molecules. In order to get a significant increase in the temperature of the air next to the ball, the air in front of the ball has to compress significantly and this will not occur unless the ball is moving at supersonic speeds.

Also, the temperature of the ball as it is passing through the air will depend on a number of other factors:

1. The mass of the ball.
2. The specific heat of the ball material.
3. The size (volume) of the ball.
   
4. The initial (muzzle) velocity of the ball.
   
5. The temperature of the ball after being fired through the gun barrel
   
6. The temperature of the air.
   
7. The pressure of the air.

AM

 

[darkdave3000]

Essentially, the loss of kinetic energy of the ball as a function of distance or time will depend on how that loss of energy is distributed between the air molecules and the ball molecules. In order to get a significant increase in the temperature of the air next to the ball, the air in front of the ball has to compress significantly and this will not occur unless the ball is moving at supersonic speeds.

Also, the temperature of the ball as it is passing through the air will depend on a number of other factors:

1. The mass of the ball.
2. The specific heat of the ball material.
3. The size (volume) of the ball.
   
4. The initial (muzzle) velocity of the ball.
   
5. The temperature of the ball after being fired through the gun barrel
   
6. The temperature of the air.
   
7. The pressure of the air.

AM

So is there a formula that has these variables you just mentioned? Then I can just input the properties and get an output per unit time.

 

[Andrew Mason]

Fluid dynamics is complicated. NASA may be able to help you as they deal with heating of spacecraft on re-entry. But these are moving much more quickly than a cannon ball.

AM

 

[darkdave3000]

Fluid dynamics is complicated. NASA may be able to help you as they deal with heating of spacecraft on re-entry. But these are moving much more quickly than a cannon ball.

AM

How about a simple subsonic model?

 

[Andrew Mason]

A golf ball driven from the tee would be similar to a cannon ball flight. Not much heating there.

AM

 

[darkdave3000]

Yes , so if you can supply a formula for thermodynamics there it would be much appreciated. Then I can scale it up to a spherical meteorite hitting the Earth's atmosphere at subsonic speeds. The drag formula already can be used to determine deceleration rate in m/s/s so that can be used to work out delta V due to drag/friction and I can thus work out total heat friction.

I can then use your formula to decide how much of that heat goes into air vs projectile and also at what rate.
The formula should be able to accommodate common materials with known properties such as aluminium.

The drag formula already accounts for the size and mass of the projectile as well as air density.

What will happen during the game/simulation is the spherical meteorite will glow red according to a chart indicating temperature in kelvin/celcius. There will also be virtual air particles glowing red according to their temperature in the trail of the projectile.

Both projectile and virtual air particles will lose heat per laws of thermodynamics/entropy.

A golf ball driven from the tee would be similar to a cannon ball flight. Not much heating there.

AM

 

[Stephen Tashi]

"Air friction" may not have much effect on a musket ball. http://thesciencepundit.blogspot.com/2010/04/air-friction-myths.html

 

[darkdave3000]

"Air friction" may not have much effect on a musket ball. http://thesciencepundit.blogspot.com/2010/04/air-friction-myths.html

I understand that but regardless, I like to model these insiginificant effects of IR radiation so that I can scale up the model to meteorites where heat does play a significant role.

 

[Andrew Mason]

What will happen during the game/simulation is the spherical meteorite will glow red according to a chart indicating temperature in kelvin/celsius. There will also be virtual air particles glowing red according to their temperature in the trail of the projectile.

Both projectile and virtual air particles will lose heat per laws of thermodynamics/entropy.

I understand that but regardless, I like to model these insiginificant effects of IR radiation so that I can scale up the model to meteorites where heat does play a significant role.

There are a lot of confused concepts here.

A sub-sonic meteorite, which would be extremely rare, would not get hot. The air around it would not get hot. The sub-sonic object would fall as any other sub-sonic object. It would reach a terminal sub-sonic velocity when the drag force became equal to its weight.

The kind of heating you are talking about occurs with supersonic objects moving at many times the speed of sound entering the atmosphere. The air does not heat up because of friction. The air in front of the object is rapidly compressed and its temperature increases due to the rapid compression. The physics of this is complicated so to determine the relationship between the temperature increase and speed of the meteorite I would suggest that you use a table of speed vs. temperature (maybe some NASA sources can help you do this) rather than a formula.

AM

 

[darkdave3000]

There are a lot of confused concepts here.

A sub-sonic meteorite, which would be extremely rare, would not get hot. The air around it would not get hot. The sub-sonic object would fall as any other sub-sonic object. It would reach a terminal sub-sonic velocity when the drag force became equal to its weight.

The kind of heating you are talking about occurs with supersonic objects moving at many times the speed of sound entering the atmosphere. The air does not heat up because of friction. The air in front of the object is rapidly compressed and its temperature increases due to the rapid compression. The physics of this is complicated so to determine the relationship between the temperature increase and speed of the meteorite I would suggest that you use a table of speed vs. temperature (maybe some NASA sources can help you do this) rather than a formula.

AM

So just to clarify, if a 1km radius meteorite traveling at mach 0.8 screams through the atmosphere it won't heat up?

 

[BvU]

https://spaceflight.nasa.gov/shuttle/reference/shutref/sts/profile.html

The entry thermal control phase is designed to keep the backface temperatures within the design limits. A constant heating rate is established until below 19,000 feet per second.

is mach 17

 

[darkdave3000]

https://spaceflight.nasa.gov/shuttle/reference/shutref/sts/profile.html is mach 17

Thanks for this clarification everyone, is it possible for you guys to supply an article link discussing the formula that would predict the heat distribution of a supersonic sphere made of some well known materials where the properties are known such as iron?

So say an Iron meteorite that is a perfect sphere screams through the atmosphere at say mach 2. What formula (regardless of complexity) should I use to enter the properties of the material and fluid and their relative speeds to each other so that I can calculate how the heat is distributed and radiated away between air and rock?

And any other related formulas such as one that predicts the varying drag coeficient at different relative speeds of the passing fluid at various densities.

I would appreciate a link to a website or a formula instead of an advice to just contact NASA as I have found it impractical to contact them.

 

[Andrew Mason]

So just to clarify, if a 1km radius meteorite traveling at mach 0.8 screams through the atmosphere it won't heat up?

That's right. No appreciable increase in temperature of the air or meteorite at subsonic speeds. This is because there is no extreme compression/shock wave produced.

Thanks for this clarification everyone, is it possible for you guys to supply an article link discussing the formula that would predict the heat distribution of a supersonic sphere made of some well known materials where the properties are known such as iron?

So say an Iron meteorite that is a perfect sphere screams through the atmosphere at say mach 2. What formula (regardless of complexity) should I use to enter the properties of the material and fluid and their relative speeds to each other so that I can calculate how the heat is distributed and radiated away between air and rock?

And any other related formulas such as one that predicts the varying drag coeficient at different relative speeds of the passing fluid at various densities.

I would appreciate a link to a website or a formula instead of an advice to just contact NASA as I have found it impractical to contact them.

The drag co-efficient does not apply at supersonic speeds and, in any event, it does not tell you what the temperature of the air will be. At very high supersonic speeds, it is the air that gets hot due to the extreme compression. Heat flow from the air to the meteorite is what raises the temperature of the meteorite surface. That air temperature is determined by the pressure of the air in the shockwave. That will depend on a number of factors including the shape of the meteorite surface. As far as I am aware, no one has ever produced a "formula" to determine this temperature.

AM

 

[darkdave3000]

That's right. No appreciable increase in temperature of the air or meteorite at subsonic speeds. This is because there is no extreme compression/shock wave produced.

The drag co-efficient does not apply at supersonic speeds and, in any event, it does not tell you what the temperature of the air will be. At very high supersonic speeds, it is the air that gets hot due to the extreme compression. Heat flow from the air to the meteorite is what raises the temperature of the meteorite surface. That air temperature is determined by the pressure of the air in the shockwave. That will depend on a number of factors including the shape of the meteorite surface. As far as I am aware, no one has ever produced a "formula" to determine this temperature.

AM

Can you confirm that the drag coefficient does not apply at supersonic speeds? I know that the drag coefficient will change at varying supersonic speeds, but youre now saying it does not apply at all? Or do you mean to just say that whatever Drag Coeficient I have for subsonic speeds is useless because they only apply to subsonic speeds?

Do you know of any formula that will predict the CHANGE IN DRAG COEFFICIENT at varying supersonic speeds?

Thanks for reminding me that air pressure and heat are related, I recall now in Chemistry about the gas laws.

I will look it up now because of this reminder. Perhaps I can use the force of drag for subsonic speeds to now calculate the average pressure on the air infront of the projectile to work out the total heat generated per unit time. Ofcourse this will only work for subsonic speeds however negligible the heat generated will be. I will then assume the heat generated remains in the air and figure out a way to predict how much of it from that fluid transfers to the skin of the projectile. Perhaps I need to read up on thermodynamics in general to find a relevant heat transfer formula.

David

 

[Andrew Mason]

Can you confirm that the drag coefficient does not apply at supersonic speeds? I know that the drag coefficient will change at varying supersonic speeds, but youre now saying it does not apply at all? Or do you mean to just say that whatever Drag Coeficient I have for subsonic speeds is useless because they only apply to subsonic speeds?

There is, of course, drag force at supersonic speeds. It is just that the drag coefficient that you have for subsonic speeds cannot be used for supersonic speeds.

Do you know of any formula that will predict the CHANGE IN DRAG COEFFICIENT at varying supersonic speeds?

I found this paper that deals with the drag coefficients for meteorites traveling at hypersonic speeds. You will notice, however, that this paper does not discuss temperature. The temperature of the highly compressed air in the shock wave is the result of compression, not friction.

AM

 

[darkdave3000]

There is, of course, drag force at supersonic speeds. It is just that the drag coefficient that you have for subsonic speeds cannot be used for supersonic speeds.

I found this paper that deals with the drag coefficients for meteorites traveling at hypersonic speeds. You will notice, however, that this paper does not discuss temperature. The temperature of the highly compressed air in the shock wave is the result of compression, not friction.

AM

Oh very nice! So I can continue to use the drag formula
https://en.wikipedia.org/wiki/Drag_equation
for supersonic speeds as long as I vary the Drag Coefficient in that formula accordingly to the supersonic speed it's at for that type of shape correct?

The paper you supplied gave some advice on how to change this Drag coefficient at various supersonic speeds.

Wikipedia does not state that the drag formula is only valid for subsonics. So can I assume it can be used for supersonic speeds as long as I alter the Drag coefficient as discussed above?

 

[Andrew Mason]

Oh very nice! So I can continue to use the drag formula
https://en.wikipedia.org/wiki/Drag_equation
for supersonic speeds as long as I vary the Drag Coefficient in that formula accordingly to the supersonic speed it's at for that type of shape correct?

The paper you supplied gave some advice on how to change this Drag coefficient at various supersonic speeds.

Wikipedia does not state that the drag formula is only valid for subsonics. So can I assume it can be used for supersonic speeds as long as I alter the Drag coefficient as discussed above?

No. There is no formula. The paper I referred you to shows a relationship between mach no. and drag for a sphere. Use that. Again, this does not tell you what the temperature of the air or projectile will be.

AM

 

[darkdave3000]

No. There is no formula. The paper I referred you to shows a relationship between mach no. and drag for a sphere. Use that. Again, this does not tell you what the temperature of the air or projectile will be.

AM

The paper tells me how to alter the drag coefficient. Youre saying it also has its own formula for drag that is different than:
https://en.wikipedia.org/wiki/Drag_equation
the above? So Drag Equation above is only for subsonic speeds? Or can I vary the drag coefficient as per paper for use with above drag formula? Wikipedia article above is valid only for subsonic? If so they should mention that and I will talk to wikipedia.

UPDATE:

re reading the paper you supplied confirms there is not drag equation, only equation to ALTER drag coefficient, so I still need a drag equation either the one mentioned above or otherwise. The output of such should be Force in Newtons due to drag.

 

[darkdave3000]

Regarding heat I will use the drag force to calculate pressure and use gas laws
https://en.wikipedia.org/wiki/Gas_laws
to calculate temperature and use
https://en.wikipedia.org/wiki/Heat_equation
https://en.wikipedia.org/wiki/Thermal_conductivity
Equations and concepts above to calculate heat distribution from air to meteorite. Let me know if this makes sense to you or not.

 

[Andrew Mason]

The paper tells me how to alter the drag coefficient. Youre saying it also has its own formula for drag that is different than:
https://en.wikipedia.org/wiki/Drag_equation
the above? So Drag Equation above is only for subsonic speeds? Or can I vary the drag coefficient as per paper for use with above drag formula? Wikipedia article above is valid only for subsonic? If so they should mention that and I will talk to wikipedia.

This NASA article explains the drag coefficient reasonably well. Note, it says:

At supersonic speeds, shock waves will be present in the flow field and we must be sure to account for the wave drag in the drag coefficient. So it is completely incorrect to measure a drag coefficient at some low speed (say 200 mph) and apply that drag coefficient at twice the speed of sound (approximately 1,400 mph, Mach = 2.0)."

The drag coefficient is used with the Drag Equation to determine the drag force. This is intended to be used for airplanes, not meteorites or re-entry spacecraft . The drag equation assumes a constant drag coefficient, which is not the case for supersonic and hypersonic (ie. extreme supersonic) speeds. For supersonic and hypersonic speeds for meteorites I would use the paper I referred you to.

UPDATE:

re reading the paper you supplied confirms there is not drag equation, only equation to ALTER drag coefficient, so I still need a drag equation either the one mentioned above or otherwise. The output of such should be Force in Newtons due to drag.

Yes. The paper deals with the drag coefficient, which I assume refers to the drag coefficient in the drag equation. You have to take the value for drag coefficient shown on the blue line on the graph on page 2 of the paper for a given speed and plug it into the drag equation to determine drag force on the meteorite. That will only give you the drag force at that speed. The drag force at a given speed gives you the rate of kinetic energy loss per unit distance traveled at that particular speed (this is different than the rate of energy loss per unit time) or the velocity loss per unit time. But, again, the rate of energy loss does not tell you what the temperature of the air in contact with the surface will be.

AM

 

[darkdave3000]

This NASA article explains the drag coefficient reasonably well. Note, it says:The drag coefficient is used with the Drag Equation to determine the drag force. This is intended to be used for airplanes, not meteorites or re-entry spacecraft . The drag equation assumes a constant drag coefficient, which is not the case for supersonic and hypersonic (ie. extreme supersonic) speeds. For supersonic and hypersonic speeds for meteorites I would use the paper I referred you to.
Yes. The paper deals with the drag coefficient, which I assume refers to the drag coefficient in the drag equation. You have to take the value for drag coefficient shown on the blue line on the graph on page 2 of the paper for a given speed and plug it into the drag equation to determine drag force on the meteorite. That will only give you the drag force at that speed. The drag force at a given speed gives you the rate of kinetic energy loss per unit distance traveled at that particular speed (this is different than the rate of energy loss per unit time) or the velocity loss per unit time. But, again, the rate of energy loss does not tell you what the temperature of the air in contact with the surface will be.

AM

Ok so I am glad we are on the same page here, I CAN use the drag formula as long as I vary the drag coefficient correctly (as described in the article you sent me) for the shape (in my case a sphere) at various supersonic speeds. Since I will now be able to know the Force due to drag I can plug in the mass of the asteroid in F = MA and work out A(acceleration). So I will know the deceleration rate of my hypothetical iron sphere meteorite, meaning that I can work out delta Velocity per unit time. Because I know that Kenergy = 0.5 X M X V^2 I can also work out the change in kinetic energy per unit time since I know the mass and delta V per unit time. That way I know change in kinetic energy per unit time and therefore the maximum amount of heat that could possibly be produced per second based on the number of joules lost per second. The question is , will 100% of that energy be heat? Or not? If not what percentage of it will be heat? This comes down to the definition of heat, is it IR radiation or just particle movement.

Once I've worked out the dilema with what heat / temperature really is between kinetic energy of air particles or IR radiation then I can use the following formulas and concepts to assist me:
https://en.wikipedia.org/wiki/Gas_laws
https://en.wikipedia.org/wiki/Heat_equation
https://en.wikipedia.org/wiki/Thermal_conductivity

I'm guessing gas laws will help me resolve my above mentioned understanding of heat, and then the following links after that will help me predict distribution of heat.

 

[Andrew Mason]

... That way I know change in kinetic energy per unit time and therefore the maximum amount of heat that could possibly be produced per second based on the number of joules lost per second. The question is , will 100% of that energy be heat? Or not? If not what percentage of it will be heat? This comes down to the definition of heat, is it IR radiation or just particle movement.

The meteorite's kinetic energy is, eventually, all dissipated in the form of thermal kinetic or potential energy of air molecules and meteorite molecules. But you cannot determine the temperature of the meteor from that.

In order to determine the temperature of the air that is in contact with the surface of the meteor, you need to know the pressure in the shock wave. That pressure is so high that the resulting temperature melts the meteorite material.. For example, if you take air at 200 K and compress it so the pressure increases by a factor of 25 times, the temperature will rise to 5000K. What you need is the pressure in the shock wave. That can be calculated from the drag force and the cross-sectional area of the meteorite. By applying the adiabatic condition ($PV^\gamma = \text{K (constant)}$ and ideal gas law ($PV=nRT$) you should get an approximate temperature from that pressure. It will be complicated to do because you need to know the initial pressure - and temperature (edit) - of the air to determine how much compression occurs.

AM

 

[darkdave3000]

The meteorite's kinetic energy is, eventually, all dissipated in the form of thermal kinetic or potential energy of air molecules and meteorite molecules. But you cannot determine the temperature of the meteor from that.

In order to determine the temperature of the air that is in contact with the surface of the meteor, you need to know the pressure in the shock wave. That pressure is so high that the resulting temperature melts the meteorite material.. For example, if you take air at 200 K and compress it so the pressure increases by a factor of 25 times, the temperature will rise to 5000K. What you need is the pressure in the shock wave. That can be calculated from the drag force and the cross-sectional area of the meteorite. By applying the adiabatic condition ($PV^\gamma = \text{K (constant)}$ and ideal gas law ($PV=nRT$) you should get an approximate temperature from that pressure. It will be complicated to do because you need to know the initial pressure of the air to determine how much compression occurs.

AM

In my software model of the Earth the initial pressure of the air is already determined with a fixed temperature of 15 degrees celcius and the pressure decreases as altitude increases per the formula for atmospheric pressure in wikipedia.

I intend to model the meteorite melting also, thanks for the above info.

 

[Andrew Mason]

In my software model of the Earth the initial pressure of the air is already determined with a fixed temperature of 15 degrees celcius and the pressure decreases as altitude increases per the formula for atmospheric pressure in wikipedia.

I intend to model the meteorite melting also, thanks for the above info.

I would simplify things a bit and create a table of values of the air and meteorite data according to altitude and then have your program access that table to do the calculations using that data:
Altitude (km) | Meteorite speed (m/s) | Air Temp. (K) | Initial air pressure (Pa) | Air Density (kg/m^3) | Radius | Mass kg. | CD factor | CD | Drag F (N)
100 | 100,000 | 190 | 100 | ___ | 2 m | 264,000 | .92 | ____ |
99 -->

etc.

AM

 

[darkdave3000]

I would simplify things a bit and create a table of values of the air and meteorite data according to altitude and then have your program access that table to do the calculations using that data:
Altitude (km) | Meteorite speed (m/s) | Air Temp. (K) | Initial air pressure (Pa) | Air Density (kg/m^3) | Radius | Mass kg. | CD factor | CD | Drag F (N)
100 | 100,000 | 190 | 100 | ___ | 2 m | 264,000 | .92 | ____ |
99 -->

etc.

AM

I am now starting to incorporate rockets and bullets into my simulator, rockets and bullets will have non spherical shapes and drag coefficients and will travel at supersonic speeds. Any suggestions on how I can calculate their drag coefficients at varying supersonic speeds?

Or are you going to suggest that I just do the same to them as what I did with the sphere according to the article but substitute the subsonic non spherical drag coefficients in place of the subsonic spherical drag coefficient of 0.47 into the same formula?

So for example a tear drop's drag coefficient is 0.04, so I double it the same mach speed the sphere's 0.47 doubles in the formula?

Lastly how do I find the drag coefficients of various rockets? ie Saturn V

 

[Andrew Mason]

I am now starting to incorporate rockets and bullets into my simulator, rockets and bullets will have non spherical shapes and drag coefficients and will travel at supersonic speeds. Any suggestions on how I can calculate their drag coefficients at varying supersonic speeds?

The paper I referred you to in my post #16 provides a graph showing drag coefficients of a sphere at subsonic and supersonic speeds. It will not necessarily work for bullets. You just take the drag coefficient from the chart and put it into the drag equation to find the drag force. For example, at Mach 2, the drag coefficient is about 1.0 and at Mach 3 it appears to be around .95 and at Mach 4, .93 and at Mach 5 and higher it flattens out to about .92. You use that number for CD in the drag equation:

$\text{Drag Force} = \frac{CD \rho v^2 A}{2}$ where $\rho$ is the density of the air, v is the speed of the meteorite and A is its cross-sectional area of the spherical meteorite ($A = \pi r^2$)

Or are you going to suggest that I just do the same to them as what I did with the sphere according to the article but substitute the subsonic non spherical drag coefficients in place of the subsonic spherical drag coefficient of 0.47 into the same formula?

So for example a tear drop's drag coefficient is 0.04, so I double it the same mach speed the sphere's 0.47 doubles in the formula?

Lastly how do I find the drag coefficients of various rockets? ie Saturn V

You can use the chart in the paper and pretend they are all spheres. Obviously this is not correct because bullets and rockets are not spheres and will not push air out of the way the same as spheres. Whether it corresponds to reality is not really important for your special effects, is it?

AM

 

[darkdave3000]

The paper I referred you to in my post #16 provides a graph showing drag coefficients of a sphere at subsonic and supersonic speeds. It will not necessarily work for bullets. You just take the drag coefficient from the chart and put it into the drag equation to find the drag force. For example, at Mach 2, the drag coefficient is about 1.0 and at Mach 3 it appears to be around .95 and at Mach 4, .93 and at Mach 5 and higher it flattens out to about .92. You use that number for CD in the drag equation:

$\text{Drag Force} = \frac{CD \rho v^2 A}{2}$ where $\rho$ is the density of the air, v is the speed of the meteorite and A is its cross-sectional area of the spherical meteorite ($A = \pi r^2$)

You can use the chart in the paper and pretend they are all spheres. Obviously this is not correct because bullets and rockets are not spheres and will not push air out of the way the same as spheres. Whether it corresponds to reality is not really important for your special effects, is it?

AM

Reality is the point of my simulator. It's a simulator not just a game.

 

[darkdave3000]

Ive started a new topic about creating my own computer wind tunnel simulator to solve this problem. If you have anything to add in there I would love to hear your thoughts.

https://www.physicsforums.com/threads/build-my-own-virtual-wind-tunnel.913428/#post-5754230

The

paper I referred you to in my post #16 provides a graph showing drag coefficients of a sphere at subsonic and supersonic speeds. It will not necessarily work for bullets. You just take the drag coefficient from the chart and put it into the drag equation to find the drag force. For example, at Mach 2, the drag coefficient is about 1.0 and at Mach 3 it appears to be around .95 and at Mach 4, .93 and at Mach 5 and higher it flattens out to about .92. You use that number for CD in the drag equation:

$\text{Drag Force} = \frac{CD \rho v^2 A}{2}$ where $\rho$ is the density of the air, v is the speed of the meteorite and A is its cross-sectional area of the spherical meteorite ($A = \pi r^2$)

You can use the chart in the paper and pretend they are all spheres. Obviously this is not correct because bullets and rockets are not spheres and will not push air out of the way the same as spheres. Whether it corresponds to reality is not really important for your special effects, is it?

AM

 

[darkdave3000]

The paper I referred you to in my post #16 provides a graph showing drag coefficients of a sphere at subsonic and supersonic speeds. It will not necessarily work for bullets. You just take the drag coefficient from the chart and put it into the drag equation to find the drag force. For example, at Mach 2, the drag coefficient is about 1.0 and at Mach 3 it appears to be around .95 and at Mach 4, .93 and at Mach 5 and higher it flattens out to about .92. You use that number for CD in the drag equation:

$\text{Drag Force} = \frac{CD \rho v^2 A}{2}$ where $\rho$ is the density of the air, v is the speed of the meteorite and A is its cross-sectional area of the spherical meteorite ($A = \pi r^2$)

You can use the chart in the paper and pretend they are all spheres. Obviously this is not correct because bullets and rockets are not spheres and will not push air out of the way the same as spheres. Whether it corresponds to reality is not really important for your special effects, is it?

AM

I wrote to some of the people who wrote that article you showed me , I haven't got a reply yet but could you have a look at this picture showing a portion of my email to them?
Here is the article again in case you need a quick reference:
http://www.lpi.usra.edu/meetings/lpsc2009/pdf/2059.pdf
Tell me what you think.