Interrelation between energy and momentum

In summary, two mine cars of equal mass are initially at rest, connected by a rope. One car is given a velocity of 4m/s, causing the rope to become taut and impart a velocity to the other car. In part A, if 40% of the kinetic energy of the first car is lost during the impact, the final velocity of the second car can be calculated using the definition of kinetic energy. In part B, the two cars are then coupled together and their final common velocity can be calculated using the momentum impulse equation. However, the solution for part A may not be accurate as it does not account for the direction of the first car after the rope goes taut.
  • #1
joemomma
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Homework Statement


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.


Homework Equations



The definition of kinetic energy:
KE = [itex]\frac{1}{2}[/itex]*m*[itex]V^{2}[/itex]

Momentum impulse equation:
[itex]m_{1}[/itex]*[itex]v_{1}[/itex]+[itex]\int\sum F[/itex] = [itex]m_{2}[/itex]*[itex]v_{2}[/itex]

The Attempt at a Solution



part B is easy, you just assume both carts are a system so there is no outside impulse.
[itex]m_{B}[/itex]*0+[itex]m_{A}[/itex]*4=[itex]m_{A}[/itex]*v+[itex]m_{B}[/itex]*v
[itex]m_{A}[/itex] = [itex]m_{B}[/itex]
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*[itex]KE_{A1}[/itex]=[itex]KE_{A2}[/itex]

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*[itex]V_{2}[/itex]-M*[itex]V_{1}[/itex] = [itex]\int\sum F[/itex]
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.
 
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  • #2
What is the direction of car A after rope goes taut?
 

What is the definition of energy?

Energy is the ability to do work, which can be in the form of motion, heat, or light. It is a fundamental concept in physics and is measured in joules (J).

What is the definition of momentum?

Momentum is a measurement of an object's motion and is the product of its mass and velocity. It is a vector quantity and is measured in kilogram meters per second (kg m/s).

How are energy and momentum related?

Energy and momentum are related through the principle of conservation of momentum. This principle states that in a closed system, the total momentum before an event must equal the total momentum after the event. This means that any changes in an object's momentum must be balanced by changes in its energy.

What is the equation for calculating energy?

The equation for calculating energy is E = mc^2, where E is energy, m is mass, and c is the speed of light. This equation, known as Einstein's famous mass-energy equivalence formula, shows that energy and mass are interchangeable and are two forms of the same concept.

How does energy and momentum affect an object's behavior?

The amount of energy and momentum an object has will affect its behavior in different ways. For example, an object with a high amount of momentum will be more difficult to stop or change its direction compared to an object with a low amount of momentum. Similarly, an object with a high amount of energy will be able to do more work or cause more changes compared to an object with a low amount of energy.

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