1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Differential Equation for deceleration of a bullet

  1. Nov 29, 2013 #1
    1. The problem statement, all variables and given/known data
    A bullet of mass m strikes an amor plate with initial velocity v0. As the bullet burrows into the plate, its motion is impeded by a frictional force which is directly proportional to the bullet's velocity. There are no other forces acting on the bullet.

    -Use Newton's Second Law to set up an IVP for the position of the bullet
    -How thick should the plate be to stop the bullet?


    2. Relevant equations

    [None given with problem]

    3. The attempt at a solution

    I attempted to use a modified version of a harmonic oscillator formula, where there is no spring. Only the momentum of the object and the friction force resisting it.

    m*d^2x/dt^2-kdx/dt=0.

    However, in solving it I've run into a problem. While I got an equation for x, x(t)=v0*e^Rt/(R-1)-v0/(R-1) (where R=m/k), trying to use that to find the time at which the bullet stops yields

    0=v0*R*e^Rt/(R-1)

    e^x=/=0, regardless of x, so there is no possible time at which dx/dt=0.

    Is my model flawed, or have I made some other error?
     
    Last edited: Nov 29, 2013
  2. jcsd
  3. Nov 29, 2013 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Think about what that equation is saying. For a greater dx/dt, what happens to the acceleration?
     
  4. Nov 29, 2013 #3
    Check your sign on the friction term ;)
     
  5. Nov 30, 2013 #4
    Acceleration becomes more negative, that is, deceleration increases with velocity. So, I could make both terms negative?

    Switching the signs in the initial equation only changes whether R is positive or negative. That doesn't solve the fact that my final equation is mathematically impossible.
     
  6. Nov 30, 2013 #5
    Yes, but making them both positive is probably less confusing. For example, the equation of motion is:
    F=ma=-kv

    Now bring them to one side, set a=dv/dt and solve for v(t). Then integrate again to solve for x(t)


    Yes, you will have a negative exponential which approaches 0 as time goes to infinity. You should be able to solve it from there.
     
  7. Nov 30, 2013 #6
    Ah, I see. So it is meant to be asymptotic? I thought it might be, but it didn't make sense from a physical interpretation, I.E that the bullet never stops moving.
     
  8. Nov 30, 2013 #7
    Yes unless I'm missing something. A better example may be a boat which turns off its motor. How far does it drift before stopping? In this case the asymptotic relation is a bit more realistic. With the bullet-wood example, the k value is much larger so it approaches 0 very quickly. It may help to think of the "wood" instead as being jello :)
     
  9. Nov 30, 2013 #8
    Yeah, that does make it a bit easier to visualize. In the case of a bullet hitting a metal plate, the medium tends to be denser than the projectile, and all sorts of deformation occurs. In real life this equation would look a bit more like this: http://i.imgur.com/2KA5jiv.gif :p


    A non-deforming projectile in a fluid medium makes the result a bit more logical. The result I ended up getting was

    xf=v0*m/k

    Seems legit.
     
  10. Nov 30, 2013 #9
    Yup that's what I calculated as well.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Differential Equation for deceleration of a bullet
  1. Differential equations (Replies: 5)

  2. Differential Equations (Replies: 1)

Loading...