# Interrelation between energy and momentum

1. Feb 19, 2012

### joemomma

1. The problem statement, all variables and given/known data
Two mine cars of equal mass are connected by a rope which is initially slack. Car A is imparted a velocity of 4m/s with car B initially at rest. When the slack is taken up the rope suffers a tension impact that imparts a velocity to car B and reduces the velocity of car A.

a. If 40% of the kinetic energy of car A is lost during the impact calculate the velocity imparted to car B.

b. Following the initial impact car B overtakes car A and the two are coupled together. Calculate their final common velocity.

2. Relevant equations

The definition of kinetic energy:
KE = $\frac{1}{2}$*m*$V^{2}$

Momentum impulse equation:
$m_{1}$*$v_{1}$+$\int\sum F$ = $m_{2}$*$v_{2}$

3. The attempt at a solution

part B is easy, you just assume both carts are a system so there is no outside impulse.
$m_{B}$*0+$m_{A}$*4=$m_{A}$*v+$m_{B}$*v
$m_{A}$ = $m_{B}$
Since they are both the same mass they each end up going 2 m/s.

Part A is confusing me. The problem states that 40 % of the kinetic energy of cart A is lost during the impact. That means that 60 % of the kinetic energy is remaining. I should be able to use the definition of kinetic energy to solve for the velocity of cart A after the impact by saying:
0.6*$KE_{A1}$=$KE_{A2}$

After that I'm not really sure what to do. I presume the change in kinetic energy can be seen as an impulse, where
M*$V_{2}$-M*$V_{1}$ = $\int\sum F$
With M being the mass of one cart, V1 being the initial speed of cart A (4 m/s), and V2 being the speed of cart A after the impact. Then that impulse would be applied to cart 2 with an initial speed of 0 m/s.

The problem is that when I take this method I solve for a post-impact speed of cart A being 3.1 m/s and cart B being 0.9 m/s. Obviously this can't be right since the stationary cart is supposed to be overtaking the cart that was originally moving.

I hope that made sense. I'm not really used to this LATEX formatting. If I can elaborate on anything please let me know.

2. Feb 19, 2012

### LawrenceC

What is the direction of car A after rope goes taut?