Maximizing Train Speed with a Damped Buffer Stop: Impulse-Momentum Approach

[izzyosman]

Homework Statement

A buffer stop at the end of a railway track has a moving part of mass, 2 Mg, which can move 2.3m parallel to the track. The force resisting the motion of the moving part is given by cxdot where xdot is the velocity and c is 200kn/m s

What is the greatest speed with which a train, of mass 100 Mg, can hit the buffer stop if, at the end of its 2.3m travel is not to exceed 1.5m/s?

Assume that after impact, the train and the moving part of the buffer stop move together.

Homework Equations

The Attempt at a Solution

I started with the impulse momentum equations

Train: -F t = 100x10^3 (1.5-x) where x is the initial velocity of the train
Buffer: (F-cxdot) t = 2x10^3 (1.5-0)

I do not know how to proceed because I have not seen an impulse momentum equation which includes a damper. Please help!

 

[izzyosman]

I've come to the conclusion that the x and xdot are the same thing as the damping force does not apply until after impact. However I also notice that i only have 2 equations and 3 unknowns: xdot, t and F, and so I'm unable to continue, can anyone think of another equation I'm missing?

 

[izzyosman]

Never mind people I got it!

Okay so what i did was start off with the integral form because the damping force changes over time:

Train: -Ft = 100 (1.5 - xdot)
Buffer: Int(F-cxdot) dt = 2(1.5-0) note that the two xdots are not necessarily the same as it changes over time.

The buffer eqn can reduce to Ft - c Int(xdot)dt = 2(1.5)
Since Int(xdot)dt = 2.3m, I now only have the initial velocity of the train, xdot, to deal with, and everything cancels out!

Yay me! :D