If physics has taught man anything other than that gravity works it would be that a projectile follows a parabolic path when launched. This is because the force of gravity is independent to the force of the combustion in the firing chamber acting on the projectile. Using physics it is possible to calculate the distance that the projectile will follow one fired. If a pressure gage is inserted into the firing chamber so that it can take a reading of the max pressure build up before the spud is propelled out of the barrel several equations can be applied and ultimately the distance traveled by the spud can be found. It is important to place the pressure gage in a fitting or thicker part of the firing chamber so that it will not be blown out of the chamber as a projectile. The preferred type of gage to be used is one that pushes a peg to the highest pressure and then stays requiring that it be reset after each use. Otherwise you would have to estimate the max pressure by inspection of the dial during firing. Once this max pressure is obtained you can use it to calculate the force on the spud in the barrel.
Where F is the force in Newtons on the spud in the barrel, p is the max pressure in the firing chamber in Pascals, and c is the cross sectional area of the spud in the barrel. To convert psi into Pascals multiply by 6.895E3. Remember that the cross sectional area of the spud is the cross sectional area of the barrel. This is found by using
Where c is the cross sectional area of the barrel, and r is the radius of the barrel in meters used.
Once the force on the spud projectile has been found Newton second law can be applied
Where force F in Newtons, is equal to mass m in kg, times acceleration a in m/s2. The mass of the potato can easily be found by finding the mass of the whole potato before being cut in the barrel and then taking the mass of that potato after being cut and to find the mass of the potato being shot out of the barrel. Set m*a equal to the above equation p*c after making the substitution for c and solve for a. This gives
a=(p*p *r2)/m (5)
After taking the mass of the projectile potato, m in kg, you will have the pressure in the chamber, p in Pascals, prior to firing the spud, the radius of the barrel, r in meters, and the mass of the potato shot. From all of the above equations it is possible to find the acceleration of the potato down the length of the barrel given by equation (4). This is of course assuming a frictionless barrel. In order to calculate the friction of the spud in the barrel it is suggested that a spud projectile of average size and density be puled out of the barrel by use of a hook through the spud and attached to a fish weighing scale. This will give the kilogram force needed to overcome fiction in the barrel but it must then be converted to Newtons by multiplying the value times g.
W is the weight read from the scale in kg, g is the gravity constant =9.81m/s2, and N is the force of the kinetic friction of the potato in the barrel in Newtons. So from the previous calculation of the kinetic fiction N of an average spud projectile it is possible to find the true acceleration of the spud as it passes down the barrel from this equation.
a=[(p*p *r2) -(W*g)]/m (7)
Where a in m/s2 is the acceleration of the spud in the barrel, p is the max pressure in the firing chamber given in Pascals, p is the constant » 3.14, r is the radius of the barrel in meters, W is the weight given by forcing an average potato out of the barrel with a scale in kilograms, and m is the mass of the potato projectile.
Where not quite done yet! We still have to find the velocity of the potato now as it leaves the barrel. This commonly referred to as muzzle velocity. This can be found from the equation
Where V is the muzzle velocity, Vo is the initial velocity before firing or 0, a is the acceleration found above, and x is the length traveled by the spud or the barrel length. This yields the following equation.
After substituting in a from equation (7) in to equation (9) You get
From this muzzle velocity V can be found by taking the square route of both sides
Once the muzzle velocity V in m/s2 is accurately calculated being careful to have the correct unit conversions the following formula can be applied to find the distance that the spud could travel if there is no air resistance or drag.
Where R is the range in meters, V is the muzzle velocity of the projectile, g is the gravity constant = 9.81 m/s2, and sin (2q ) is the sin of 2 times the launch angle q . When (11( is substituted into (12) the following equation results.
Again to summarize:
V Muzzle velocity (m/s)
x length of barrel in (m)
p Max firing chamber pressure in Pascals or (N/m2)
p » 3.14
r radius of barrel in (m)
W weight shown on scale when pulling spud out of barrel (kg)
g gravity constant = 9.81 (m/s2)
m mass of the spud projectile in (kg)
R is the range of the projectile in (m)
q is the launch angle
R is the range of the projectile in (m)