Reloading: E= and all that Stuff

This month we’re doing a bit of revision, just in time for your end of term reloading exams. Calculators, pencils and paper will be needed. The subjects are energy, gravity, acceleration, recoil, drag, rotation of the earth, centripetal and centrifugal force.

Energy – In Both Directions

Let’s start with a trio of easy ones. Free Recoil Energy is the figure that denotes how hard the rifle is going to hit you when you pull the trigger. There are several versions of the equation, all employing K numbers (constants) as fiddle factors. The one published by Lyman is my favourite:- (Equation 1)

Where W1 is the weight of the bullet in pounds, W2 is the weight of the powder charge in pounds, Wg is the weight of the gun in pounds and Vp is the muzzle velocity of the bullet in fps. As an example, take a 200-grain pill, divide by 7000 (number of grains in a pound) and we get 0.029. The 50-grain powder charge divided by 7000 gives us 0.007. Take an 8 lb rifle and an M.V. of 3000 fps. (Equation 2)

That equals 27.93 ft/lbs. I’ll go with that!

 

Foot Pounds

At the other end of the rifle we can calculate the muzzle energy. The equation uses a standard figure for the constant which can be refined if you know your elevation and the true figure for g.
The calculation uses 32.17ft/sec/sec. (Equation 3)

Taking the sample numbers from our FRE calculation we get (Equation 4)

That equals 3996 ft/lbs Muzzle Energy. Overkill for the local muntjac!

Now to SD, Sectional Density. The bullet weight in pounds divided by the square of its diameter in inches. So, using the sample numbers from our 300 Remington Ultra-Mag above we get 200 (divided by 7000) equalling 0.029 divided by .308². That’s 0.029 over 0.094864. That gives us an SD of .306. Lyman quote .301 but that’s close enough.

 

It was Bernoulli

The family Bernoulli are probably one of the most important that you’ve never heard of. Generations of them came up with awesome inventions, mathematical equations and other chunks of new science. Our star is Daniel Bernoulli, born in the Netherlands but forced to move to Switzerland to evade religious persecution. Amongst his equations (Bernoulli’s Principle) are a couple that identify and explain the behaviour of a projectile in air. The first is good old drag. We get to R∝v² . That is, Drag (R) is proportional to the square of Speed (V). A simple example is to compare two identical bullets, one travelling at 1X fps and the other at 2X fps. If we ignore the effect of gravity (drop) then it would seem reasonable to assume that the 2X bullet would travel twice as far as the other. Not so, there will be a significant shortfall due to drag. In other words, doubling the distance that a bullet is required to travel requires much more than a doubling of the muzzle velocity. In the real world we have to add gravity to create a notionally parabolic trajectory, but the principle of the rule still applies.

 

Low Drag Saves Fuel!

If we fired any projectile in a vacuum (on earth) then its shape would have no impact on its flight characteristics – its trajectory merely being a function of speed and gravity.

However, the presence of our atmosphere creates a serious complication. Drag changes trajectory and performance and it’s this element that plays the major part in calculating the all important Ballistic Coefficient (BC). The sleeker the better, but strangely, not just the front. A tear-drop section with a pointed tail would be optimum, but a sod to load and fire! The bigger the number, the better the BC! Almost all loading manuals quote a BC figure for each bullet design, ranging from 0.050 to 0.560. This can be used in conjunction with its muzzle velocity to calculate the trajectory and from that, the maximum practical range for accurate use. As an example let’s take a look at a couple of 165-grain Speer .308 Spitzer bullets. The square base has a BC of 0.433 and from an MV of 2700 has a residual velocity of 1747 fps at 500 yards. The sleeker Boat Tail version has a BC of 0.477 and from the same MV is still travelling at 1834 fps at 500 yards. We have (Equation 5) Where SD equals sectional density, an equation we examined earlier, i is the Form Factor, a standard model number calculated by dividing the drag coefficient of the actual bullet by the standard or ‘reference’ GI bullet model. Simplifying the fractional equation we get BC equal to M (Mass in lb or kg) divided by i times the diameter squared (in either inches or metres). Because of the presence of a ‘factor’, this model is exclusive to ballistic calculations for bullets.

 

Bend It Like Beckham

Bernoulli is also credited with the maths to explain lift, as in the wing of an aeroplane. This brings us to another couple of aerodynamic effects on our speeding and spinning bullet. The effect of the spin is dynamic stability, just like a child top, an artificial horizon or the wheels of a motor cycle. It is a desirable effect the first examples, but makes it harder to change direction on a motor cycle as the speed increases. Throughout its flight, the imparted spin therefore tries to keep the bullet pointing in the same direction, at an increasing deflection angle to its’ trajectory. The ever increasing ‘nose-up’ effect causes lift and, in extremis, tipping or tumbling. The rotation of the bullet also creates boundary drag around its calibre and as the tipping process begins we get a secondary ‘lift’ called the Magnus Effect. It is small but measurable and induces a tiny parabolic curve into the trajectory. It is better visually illustrated by the effect of spin on a tennis ball.

 

Angle Of Dangle

If you’re asked to estimate from which muzzle angle does the bullet travel the furthest distance, the obvious answer would be 45º. However, if we apply drag and g (gravity) to our distance calculation we get another change to the ‘obvious’. We have to assume that our bullet trajectory is within the bounds of our atmosphere rather than ballistic for the figures to work, but the result is actually an angle of 32º to the horizontal.

The reason is actually pretty obvious, since g is acting upon the bullet at 32.17 ft/sec/sec (9.805 m/sec/sec) from the moment it escapes the muzzle. As the M.V. decays so the ratio of linear travel to descent due to gravity changes. More modified parabolas.