Explosion Speed Calculations
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The timeframe in which an explosion, which was caused by an explosive, expands a certain distance can be used in order to figure out a timeframe used to calculate the speed of a character.
In which cases can it be used?
In order to use this method to determine a timeframe the explosion has to be caused by a explosive, in other words by a reactive substance that contains a great amount of potential energy that can produce an explosion if released suddenly, usually accompanied by the production of light, heat, sound, and pressure.
That means that explosions caused through things like supernatural powers, lasers, lightning etc. can not be calculated using this method.
In order to get a result one will require a way to figure out the weight of the explosive used, determine or make a reasonable guess on the type of explosive used and figure out how far the fireball or blast wave of the explosion expanded in the timeframe.
When it is known how far the character moved relative to the explosion
Step 1: Making a guess on the type of explosive
The typical types of explosives one would see in media are C4 and TNT (Trinitrotoluene).
C4 is a type of plastic explosive, which can be freely formed like clay, usually comes in cuboid shaped packages and requires a detonator to explode.
If C4 isn't stated or something very similar to it is described or shown it usually is best to assume TNT is used.
Step 2: Determining weight of the explosive used
The weight/mass we determine through the following methods will be our "charge weight".
Method 1: Volume and density
If the explosive (bomb, or similar) is visible this method can be used. For it one first has to scale the volume of the explosive.
In order to then figure out the mass one has to multiply the density of the explosive with the volume.
Usual density of TNT is 1.654 g/cm^{3} ^{[1]}
Usual density of C4 is 1.72658 g/cm^{3} ^{[2]}
One should be careful to use the correct dimensions.
Method 2: Typical sizes and density
Explosives might appear in characteristic sizes.
A package C4 for example typically is 2 inches by 1.5 inches and 11 inches long, weighing 1.25 lb (0.57 kg). ^{[2]}
If only the typical volume, but not the typical weight is known one can figure out the weight using the density, like in method 1.
Method 3: Relation to fireball radius
This is the least secure method and should only be used if the other two aren't possible.
The relation between fireball size and weight is:
D = 8.5 x W^{0.341}
Where W is the weight of the explosive in pounds and D is the diameter of the fireball in feet. ^{[3]}
If this method is used the type of explosive should be assumed to be TNT.
Step 3: Scaling radius of explosion
Next we want to figure out the radius the explosion extended to during our timeframe. Note that the timeframe starts at the point the explosion starts and not sooner.
In order to do so we just use the usual scaling methods.
One should pay attention that we are strictly speaking scaling how far the shockwave expanded, not the fireball. So if possible the extension of the shockwave should be estimated. Since that often is not feasible for short distances it is assumed that the fireball extents with the shockwave, if we talk about fireballs at larger expansion distances this can not be assumed anymore.
The result of the scaling of the radius is our "range".
Step 4: Figuring out the timeframe
For finally getting the timeframe we will use this calculator.
For the "Explosive Type" we will choose the one we guessed in Step 1.
For the "Charge Weight (kg)" value we will use the weight/mass estimated in Step 2. Keep in mind to convert it to kilogram before use.
One thing that might be necessary here is to convert "free-air data" to "surface data". The calculator calculates the explosions in terms of an explosion at the surface, which means a hemispherical explosion which at one side is restricted by the ground. Because of that if an explosion happens in the air or generally expands in all directions without being restricted at one side, the weight of the explosive has the be halved in order to account for that.^{[4]}
For the "Range (m)" value we will use the explosion radius determined in Step 3. Keep in mind to convert it to meter before use.
Then we click "Calculate Blast Parameters" and get a list of calculated values.
The only value interesting for us is "Time of Arrival (ms)". That value is the timeframe we wanted to determine. Using that one can calculate the speed of things that happened in the time the explosion expanded to the used range. Take heed that the value is in milliseconds (ms) and has to be converted to seconds before in speed in m/s can be calculated.
1 ms = 0.001 s
When it is unknown how far the character moved relative to the explosion
This calculator can be used in order to get a result. To explain which are the inputs have to be and what the outputs mean:
The Start Distance is the distance the character has to the origin of the explosion at the point in time the explosion happens in. The value has to be input in meter.
The Charge Weight is the weight of the explosive. See Step 2 of the prior method on how to determine this. The value has to be input in kilogram. The Degree of Derivation from Expansion Direction is an angle input in radians. This angle is there in order to describe in which direction the character dodged. If you imagine a line between the point the character starts at and the origin of the explosion, then 0 would mean that the character runs further into the direction of that line away from the origin of the explosion. If the character takes another path, then the Degree of Derivation is the angle between the 0 path and the path the character actually takes. So fro example running straight towards the explosion is π.
The Explosive Type is the type of Explosive used. This can be figured out as in Step 1 of the prior method.
The "Escapes Shockwave until extended" value is the value of how far the shockwave extended until it caught up to the character or until it was stopped from extending. In other words It is the values on how far the character had to move in order to completely have escaped the shockwave. If the shockwave was escaped completely and not restricted in its expansion this field can be left blank.
The "Speed to outrun Explosion" value is the speed in m/s necessary to escape the shockwave of the explosion. It is the main result we want.
The "Time of arrival at Start Distance" is the time it takes the Shockwave to reach the Start Distance that was input. The unit is given in ms.
The "Closest to Shockfront at" value is a control value. It specifies at which distance from the origin of the explosion the shockwave catches up closest to the character when the character travels with the "Speed to outrun Explosion". It is given in meters.
Known flaws: Things one has to keep in mind when using
- The pi error: When the Degree of Derivation gets closer to pi the speed to outrun the explosion the program gives becomes more inaccurate. To be specific it becomes smaller than it should be. The Reason for that can be easily seen if on looks at the case of the Degree of Derivation being chosen as pi. In this case the character would in the scenario run towards the origin of the explosion. It should be absolutely impossible to outrun the explosion this way, so we would expect the program to result in an error or give infinite as speed necessary to outrun the explosion. In practice you will only get a very high value though. That is due to the fact that the program only checks whether the shockwave is further away from the origin than the character at 0.1m intervals. Essentially the speed value one gets in the result here, will be the one high enough so that in one check the character is still in front of the shockwave and at the next check the character has moved all the way to the other side of the of the explosion and shockwave. In reality that would mean that the character ran through the explosion and got hit, but since the program doesn't check the characters position during that movement (if the speed if high enough) it doesn't notice that.
- Not outrunning the explosion: Take note that the character is said to have outrun the explosion if it left the range from the explosion origin in which the shockwave can be correctly approximated. That means for very large Start Distance the character might has to only move a very short distance compared to the shockwave. If the value to outrun the explosion becomes lower than the speed of sound, this is likely the reason.
- What if only the fireball, but not the shockwave was escaped from?: In that case the "Closest to Shockfront at" value becomes important. As long as that value is smaller than the radius of the fireball, the estimation is still correct. Otherwise set the "Escapes Shockwave until extended" value to the radius of the fireball.
References
- ↑ https://en.wikipedia.org/wiki/Trinitrotoluene
- ↑ ^{2.0} ^{2.1} https://en.wikipedia.org/wiki/C-4_%28explosive%29
- ↑ http://www.boomershoot.org/general/Effects.htm#Fireball
- ↑ https://books.google.de/books?id=8E6fVA6qrVkC&pg=PA62&lpg=PA62&dq=peak+side+on+pressure+tnt&source=bl&ots=uDFj7kSgJB&sig=nUzN7bqCo313rsw1sRlBqN0XkbA&hl=de&sa=X&ved=0ahUKEwjDwo6W-uPMAhWDtxoKHaS2DVcQ6AEIKjAD#v=onepage&q=peak%20side%20on%20pressure%20tnt&f=false