Ballistics of Flying Rock

The rocks could have left the blast at the same velocity but at different angles.

We have a open cast mine next to our suburb (they have thrown us with some rocks before) and they take a high speed video of each blast from which the velocity etc can be determined.

Why do you need the velocity?

You would have to know the height achieved or angle of launch....

http://www.physicsclassroom.com/class/vectors/Lesson-2/Horizontally-Launched-Projectiles-Problem-Solving

There is an infinite number of solutions.

For each angle of initial projection, there is one velocity and only one, that gives the required distance of projection.

Assume a most likely angle value, say between 30 degrees and 60 degrees, and calculate. This will give a range of around 2:1 in the velocities.

If the rocks are substantial enough to ignore air resistance then Newton can, with his equations of motion. It becomes O-level Physics.

The drag at supersonic speed need to be taken into account.

I doubt that supersonic is possible under the circumstances, but am not sure.

The dolomite mine next to our suburb launched some rocks at us (700 m away) and they made the high speed video and their calculations available. The initial speed was indeed supersonic.

Usually in an explosion (one that either accidentally or intentionally used explosives), at least about 99.99999999% of the time, there is going to be a shock wave, meaning that the gases are moving in excess of Mach 1. Another way to model the advancing gas front is by the pressure wave, i.e. - the maximum level of pressure reached, and using that as an active force upon items in the path of the shock wave, but then one would need to include something in the model for how this pressure wave dissipates over time (and distance). For a release of compressed gas, there is not necessarily a shock wave, but the similar principle applies.

For instance, a 12 Kg rock with S.G of 3.1 (some of the denser rock types), would require an initial pressure of 44Kpa, with a time constant of 2.85 seconds, thus the pressure decay integral (exponential assumption) is 15,438 Kg-m/sec³, the impulse integral becomes 460.1 Kg-m/sec (momentum), and the velocity is 38.34463, which will give a range of 150.00m when launched at a 45° angle.

The starting velocity is zero feet per second.

If starting velocity is zero, then the range is zero.

At the beginning of the explosion impulse event, all of the rocks must be presumed to be a zero velocity as a boundary value condition. Then you input the mass and area of the rocks (that will be exposed to the shock wave), and with a multi-physics model (high dimensional order of calculation), one can propose various other boundary conditions as to the the brissance of the explosion (pressure and duration of the shock wave), various likely angles of launch, etc. and develop debris patterns. At some point in the modeling, you will end up with something resembling an answer.

Yes, that is the velocity at t-0, what he is looking for is the velocity at t+0, when the energy from the explosion creates a 'step input' in the equation (actually a steep ramp/curve, but the math works easier with a square wave style input, and the results are 'close enough' to right for a real-world modeling exercise. (If you want 'full accuracy, you also have to account for things like the gravitational effects of Pluto, Alpha Centari, and the Horse Head Nebula.Remember, there is no known way to 'shield' from the effects of gravity, and gravity has no practical limit on its range [although, since we have determined that 'gravity waves' travel at the speed of light, any object so distant that the expansion of space has the object effectively 'accelerating' from us at 'superluminal' speed will have no effect, but then we won't know about those objects, since the LIGHT from them will never reach us either.])

This guy looks legit, I don't see the need to throw out the 'CR4 is not your homework hotline' responses.

Not enough data. If you can come up with something to eliminate factors that revert from normal distribution of angles and velocities, you can come to some conclusions, but it will be only pointers, not real data, since the stone number is too small to assume that any systematic errors batch get any self-canceling. On second thought, the only people for the job are the mathematicians that devised the real world models that gave "proof" of AGW . They can calculate each and every stone speed, angle, weight, content and shape, and can give you computer generated photos of every stone, just from your data, and then force the people that are responsible for those flying stones to pay the tax they calculated for them, because I'm sure from your data they can even tell who threw them. (LOL) S.M.

Cheers, going out today to document the fly rock then its office work to get some answers. I'll try these eq's. Cheers

Watch nearly any sporting event that uses a ball. The variety of paths the ball can take to end up in the same spot is almost endless.

Now there is a minimum velocity that your rock has to have started with to reach your target. This can be found by calculating the energy and momentum needed to travel your distance with a 45° angle above a level plane. Fortunately for you, the shape, size, rock type and weight will have little difference on the required launch velocity until one approaches the terminal velocity or the speed of sound in air of the rock in question, which ever is lower. Unfortunately for you many unknowns like a difference in elevation, curve of the Earth, blocked trajectory path, mid-flight collision and possible mid-flight separation can dramatically change the minimum velocity required for a completely air-borne move. I almost forgot to mention tumbling after impact can also change your minimum velocity based on angle.

That should be enough of an explanation with a refresher of your Physics 101 class that you can make a reasonable calculation with known uncertainties.

Starting velocity is '0'.

There are Excel spreadsheet programs. Google "Ballistics excel spreadsheet programs"

but the shape my have a effect, but could be nominal

may help...

The dinosaurs were laying eggs millions of years before the first birds evolved to become chickens, so obviously the egg came first.

Dang, now you've got me thinking about bacon too.

I wonder who discovered bacon?

The Pig Whisperer...

Because when you drill down to find the answer about anything...... it always come down to bacon......

mmmmmmmmmmm BACON...... ^{said in a homer simpson voice.}

Moi?

--Miss Piggy

Roger

I think in this situation, you still need a boundary value at t=0 of v=0. Then you have to have an impulse model to accelerate the object(s). Knowing this is a cave collapse helps, because now you can model the pressure wave based on air volume in initial cave, height of the collapse, and any knowledge about the entrance to the cave (probably where flying rocks are propelled from). Having that, one would calculate the approximate time duration of the collapse, apply a model to describe the decay of the pressure wave, hence arriving at the impulse (and hence the momentum change and terminal velocity of the hurled objects), neglecting air friction for the propelled objects, and assuming they do not roll when the strike the ground.

This starts to become highly complicated if roll travel is introduced. Even at the simplification level I used in my spreadsheet, there is a lot of room for error in angles, terminal velocities reached, etc.

At this time, it is plausible the subject rock was on the surface, and was carried along with the mass ejection of a blowout and not expelled from the cave at all.

The subject rock may have gone to a thousand meters altitude.

If there is a change in speed, direction, or both, then the object has a changing velocity

Check for gravitational and magnetic anomalies for accuracy.

Score one for beyond comprehension of accuracy

Thanks for the replies.

A bit more information, These fly rocks were ejected from a collapsed cave so no explosives were used just a compressed air blast. Snapped a ten inch thick tree in half around 100m away and i have since found a rock weighing about 50kg (110pd) approx. 250m (273yd) away.

Basically I've been further asked by my boss to create a heat plot of the potential velocities calculated from the fly rock and thus risk potentials. Not sure how I'm going to achieve this but I also need to figure out the likely wind speed to launch a variety of rocks and snap a ten inch tree in half.

Pictures, soil type, rock class (sedimentary/igneous), mineral type, size of cave, size of hole, any other vents for compressed air.

No chance of a gas explosion? Vandalism? Evidence of heat from air compression?

In other words, a report on your findings would help a LOT.

I'm having a problem getting my head around enough force to blow down trees 100m away. Any evidence of rock damage to the tree?

Usbport. Come on, work with us. The rock's trip starts from zero FPM and ends at zero m/s. I love mixing units that I don't understand. That's one hell-of-a-lot of compressed air, and heat.

"I'm having a problem getting my head around enough force to blow down trees 100m away."

Agreed. Together with a 50kg rock ejected to 250m. I have a mental image of this irregularly shaped rock being shot from a barrel with a diameter of, say, 10 times the rough diameter of the rock. Wildly approximates a single toy marble being blown from a 4" sewer pipe, at an upward angle. That's a LOT of air!

You really got into the dance here...GA, I think..

As simple-minded as I am, I took the Range Equation and ran with it on an Excel spreadsheet.

Assuming you can at least measure things like rock density, approximate all rocks as "spheroidal", apply a starting pressure, with an exponentially decaying pressure as the "shock" wave (which will not be > Mach 1 in this case), as long as you input the mass, and correct cross-sectional area of the rock, and calculate for a launch angle of 45 degrees, and apply a series of different pressures and decay constants, you will arrive at a series of ranges for each rock, as long as the 45 degree trajectory does not fall short of the observed range, you can back calculate the actual angle of trajectory for that rock at that pressure/decay constant. It is not a difficult calculation at all to set up and at least arrive at some initial approximations. You may be able to arrive at a "decay constant" based on the size of the initial cave, thus calculating by gravitational acceleration the time required for the "piston" to fall to the bottom.

This smacks of IED levels and the pressure waves associated with them. just saying....

The depth and angle of penetration into the ground might help identify the direction from which the rock came, and it's speed.

Does the emerging cave passage have a reasonably defined angle from horizontal? (This would narrow down the trajectory versus rocks going every which way.)

You could borrow a 3D scanner from CSI and get a plot to work with and track back.

I found this inscribed on a rock that narrowly missed me:

Hello Rocdoc26, Happy Wednesday. I'm an electrical engineer but I love solving problems. I'm a new member here at IHS and expect to get flamed but here goes, this should be fun...

As discussion enhancement, I think there are a few information points which will be needed to reach a relatively accurate conclusion. How accurate do you want your velocity calculations to be?

While pondering your question in the moment, it sounds like the rocks are given propulsion via an exothermic explosion. If true, information about the explosive/propellent characteristics could perhaps offer a baseline to begin simulations, what is the reaction speed and energy released by the explosive?

What is the angle of trajectory? The parabolic arch of flight can greatly affect the distance of travel.

The rock shape and rotation becomes a question. Shape will effect resistances due to wind and pressure. Angular momentum will certainly effect flight distance, consider a bullet fired with and without rifling.

What was the elevation from sea level, temperature and barometric pressure? Resistance and friction coefficients will have some effect on flight characteristics.

Did all the rocks fly together? Impacts would effect trajectories and speeds.

(Flame Time!) Wikipedia (cough) has some effective equations to start calculations and a decent discussion under the tag "Trajectory of a Projectile" found at http://en.wikipedia.org/wiki/Trajectory_of_a_projectile

At first glance, as more data would be helpful and when known precision will be increased. Perhaps I've been helpful and perhaps not, it was fun to post and I wish you a fantastic afternoon. Good luck with your research.

If you want the minimum starting velocity, corresponding to launch angle 45° which gives maximum distance, it's √(g*D) as included in other posts. That excludes air resistance but should be near enough.

The actual velocity could be higher than minimum, depending on the launch angle. For angle θ velocity = √(g*D/sin(2*θ))