Equations used:
Force from Pneumatic cylinder (outstroke)
F =P* π*d^2/4
Where P is the given air pressure in the lab
d is the full bore piston diameter

Work done to mass by pneumatic cylinder is given by
 W=F*x
Where x is the piston stroke length



The gravitational potential energy of the projectile and clamp system is given by
 E_GPE=m*g*h

H is the vertical distance the mass drops
M is the Projectile and clamp system mass


The energy of the projectile when it strikes the guard is given by
E_(TOTAL )=E_GPE+W
Explanation of values used:
A range of full bore piston diameters and stroke lengths were explored.
A smaller stroke length was desirable as it reduced the likelihood of the piston rod buckling which can occur in high energy impacts. This then required a large diameter for the full bore piston. A 340mm full bore piston diameter was chosen as this was an industry standard size. To calculate the stroke length the energy required from the pneumatic cylinder was calculated.  This was done by finding the difference between the total required energy and the gravitational potential energy of the projectile. The energy required from the pneumatic cylinder was then divided by the force applied from the pneumatic cylinder to find the stroke length which was 0.15.  The diameter of the piston rod was chosen in relation to the diameter of the full bore. Similarly to the full bore piston diameter the piston diameter was an industry standard size of 240mm
A relatively large height was chosen for the projectile to be held from as this gave sufficient room and time for the brakes to function in the case of an aborted test. To reduce the mass of this clamp system it was made from aluminium rather than steel.

Actual calculations:
F=6 × 〖10〗^5* π*  〖0.32〗^2/4=48254 N
				W=48254 * 0.15 = 7238J
                                                          E_GPE=(45+45.9)*9.81*3.3 = 2943

E_(TOTAL )=7238+2943=10181J

This is the theoretical maximum energy that the mass can hit the guard with however due to various losses this will never be achieved. Losses may range from 10%-5% which results in actual energy of the mass to be 9163J - 9672J respectively. This is within the desired range of 8000J-10000J. The linear encoders at the bottom of the guide rails will check the speed of the p to confirm the exact energy of the mass.

Assumptions:
The main assumption made in these calculations will be a 10% loss in energy due to various factors. This includes friction between the guide rails and carriage as well as losses in the cylinder itself. Efforts to reduce the losses in the cylinder include the use of nose bearings.
Another assumption was that the negligible difference in full bore piston area caused by the piston rod bolt was not accounted for in the pneumatic cylinder force calculations.
It was assumed that all the potential energy of the clamp system went into the final projectile energy as it strikes the guard.
The piston rod was treated as a simple cylinder however; it had a hexagonal section to help secure the piston.
It was also assumed that the weight of the cylinder rod did not affect the total energy given to the mass. This is because as the piston is forced out some of the applied energy is used to overcome the air in the bottom of the cylinder (which was supporting the weight of the piston rod). * might not need this