Hi all,


I'm having trouble working out how much energy will have to be dissipated in a test rig for a nail gun.


From previous testing, I know that a test slug fired from the gun has a mean energy of 100J.


I now want to design a test rig which can fire the gun repeatedly for 100,000 shots.


I have decided that firing a test slug vertically upwards in to the bottom of a heavy piston in a steel tube will be the most suitable design.


Each time the slug is fired from the gun, it collides with a rubber-bottomed piston in a steel tube, thus transferring its energy to the piston by pushing it up the tube. The piston can then bounce up and down in the tube until its energy is entirely dissipated as heat and sound. The process is then repeated 100,000 times.


The problem I have with designing the rig, is working out how high the piston will travel within the tube. I know I'm making a flawed assumption somewhere so please let me know where my flawed reasoning is!


Here is my working and reasoning:


Assumptions:
- neglect energy dissipation due to air drag
- since the rubber bottom of the piston has a very high spring constant, assume that once the piston and slug collide, the slug and the piston both travel with the same velocity


m_s = mass of the test slug = 0.09 kg
m_p = mass of the piston = 10 kg
E_i = energy of the test slug exiting the gun = 100 J
E_f = combined kinetic energy of the test slug and piston immediately after the collision

v_s = velocity of the test slug before the collision
v_f = velocity of both the slug and the piston immediately after the collision


Working out the velocity of the test slug:


v_s = sqrt(2*E_i / m_s) (from the kinetic energy equation)
= sqrt(2*100 / 0.09)
= 47.1 m/s


Using the conservation of momentum principle, determine the velocity of the slug and piston immediately after the collision:


m_s * v_s = (m_s + m_p) * v_f
v_f = m_s/(m_s + m_p) * v_s
= 0.09/(0.09 + 10) * 47.1
= 0.42 m/s


Combined kinetic energy of the slug and piston immediately after the collision:


E_f = 0.5 * (m_s + m_p) * v_f^2
= 0.5 * 10.09 * 0.42^2
= 0.9 J


Thus the maximum height that the piston will reach:


h = E_f / ((m_s + m_p) * g)
= 0.9 / (10.09 * 9.81)
= 9 mm


I then double checked the maximum height of the piston using the conservation of energy equation and the initial energy of the test slug. I assumed that the kinetic energy of the test slug would be entirely converted to gravitational potential energy of the piston once the piston reached its highest point. Thus energy losses due to sound and drag etc were neglected.


h = E_i / ((m_s + m_p) * g)
= 100 / (10.09 * 9.81)
= 1 metre


Thus the maximum height differs significantly depending on the calculation method I use.


Is my conservation of momentum approach correct?
Is my conservation of energy approach correct?


My intuition tells me that the conservation of momentum approach is correct, however, I'm suspicious since I do not believe that the kinetic energy of the system would be reduced from 100J to 0.9J from the collision (i.e. huge energy losses to sound and heat).