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Piston/bullet inelastic collision (not like other problems)

  1. Jul 21, 2015 #1
    1. The problem statement, all variables and given/known data
    a bullet has mass=0.01kg velocity=150m/s, you have a cylinder with 5.0m^3 of air at STP with a piston at one end that has mass=0.1kg and surface area of 10cm^3 and is at rest. The bullet strikes and embeds itself into the piston.
    The system is to be considered adiabatic after the collision

    a)find the displacement of the piston

    b)what is the time period of oscillation of the piston

    2. Relevant equations
    momentum, p=mv momentum of an inelastic collision= m1v1+m2v2= (m1+m2)V(final)
    kinetic energy of the system before collision=(1/2)[mass(bullet)][velocity(bullet)]^2
    kinetic energy of the system after collision=(1/2)[mass(bullet)+mass(piston)][velocity(final)]^2
    F=ma, F=kx
    time period of linear oscillation= 2(pi)sqrt([mass(bullet)+mass(piston)]/k)

    3. The attempt at a solution
    I calculated the final velocity- V(final)=[0.1*150]/[0.1+0.01]= 13.63636363...m/s

    I calculated the kinetic energy before- (1/2)[0.1][150]^2=112.5 joules
    I calculated the kinetic energy after-(1/2)[0.1+0.01][13.636363...]^2=10.2272 joules
    Kinetic Energy lost=102.2728 joules <-this energy lost is imparted to the gas in the cylinder as heat(that is what we are told)
    I know that a=(kx)/m
    omega-w=sqrt(k/m) there for a=sqrt(k/m)[x/m]

    I know that the gas in the cylinder exerts a force on the piston and this force will slow the pistons movement till it stops and then the force will over come it and push the piston out till a "vacuum" force pulls the piston back in and thus the linear oscillation begins. I can't for the life of me figure out how to find "x" or "k" for that matter.

    The only other hint we were given is that air is to be considered diatomic for this problem and thus Cv=5/2 R and Cp=7/2 R thus gamma can be calculated as 7/5
    but I don't know how this plays into it. Any insight would be helpful I have been staring at this and googling for at least 8 hours now.
     
  2. jcsd
  3. Jul 21, 2015 #2

    SteamKing

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    You have a piston and a cylinder filled with air. The bullet strikes the piston and drives it into the cylinder. What happens to the air inside the cylinder?

    The clue is in this statement, "The system is to be considered adiabatic after the collision." What's the relationship between pressure and volume in a cylinder which can be considered adiabatic, i.e., there is no heat transfer to or from the cylinder to its surroundings?

    https://en.wikipedia.org/wiki/Adiabatic_process
     
  4. Jul 21, 2015 #3
    Consider P when the piston is displaced by some distance x . At that instant , what is the pressure ?

    You will get P as a function of x , and thus F as a function of x .

    So use a kinematic equation → a = v*(dv/dx) and solve from there .

    Hope this helps .
     
  5. Jul 21, 2015 #4
    Okay this helped alot... I can see that 6e0bc8d5bc392b6a685e362884024348.png
    This would allow me to find P2 and thus V2, V2-V1 would tell me the change to give me x

    The part that I'm stuck on is how to find T2. I know that I have 102.2728 joules of energy creating the temperature change thus I would be tempted to use Q(heat energy in joules)=mcΔT, but air's specific heat capacity changes with temperature and pressure thus how can I calculate delta T without knowing the P2 at which the specific heat capacity would be applied. Perhaps, I should just assume that the specific heat capacity is 1kJ/kg*K since the changes to the specific heat are rather small. Sorry for the questions. I think I might be over thinking this...
     
  6. Jul 21, 2015 #5
    Do not use relation between P and T ; use the relation between P and V .
     
  7. Jul 21, 2015 #6
    The only other equation I know is ((P1V1)^gamma)/T1=((P2V2)^gamma)/T2) this was provided by our professor because he said you have to account for the temp change but this still leaves me with too many unknowns.
     
  8. Jul 21, 2015 #7
    The basic equation for an adiabetic process is P*(V)∧γ = constant .
     
  9. Jul 21, 2015 #8
    Are you sure about this equation ?
     
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