Show momentum is conserved in two different frames (relativity)

In summary, the 2000-kg car moves with a speed of 20 m/s and collides with and sticks to the 1500-kg car at rest. Because momentum is conserved in the rest frame, momentum is also conserved in the reference frame moving with a speed of 10 m/s in the direction of the moving car.
  • #1
Ravenatic20
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Homework Statement


A 2000-kg car moving with a speed of 20 m/s collides with and sticks to a 1500-kg car at rest. Show that because momentum is conserved in the rest frame, momentum is also conserved in a reference frame moving with a speed of 10 m/s in the direction of the moving car.

Homework Equations


Not sure

The Attempt at a Solution


Let's have the larger (2000-kg) car be mass M, and the smaller (1500-kg) car to be mass m. Car M is traveling at speed v. After the collision, the two cars become one mass (M+m) and its velocity we will call v'.

To an observer on the ground...
mv + 0 = (M+m) v'
v' = MV/(M+m)

To an observer in a moving frame...
M is moving at speed V-v (towards the smaller car, m) and m is moving at speed -v (towards the larger vehicle, M). After the collision, (M+m) is moving at speed v'-v.
M(V-v) - mv = (M+m)(v'-v)
MV - Mv - mv + Mv + mv = (M+m)v'
v' = MV/(M+m)

These two equations are the same, meaning the final speed of the indecent is v' from any observer. Does this mean momentum is also conserved in a reference frame? If I'm on the right track, what good would it do plugging in numbers?
 
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  • #2
Re-read the question. You need to prove that conservation of momentum in the rest frame implies conservation of momentum in the moving frame. You're also not being consistent with your notation. The v in your first set of equations represents the speed of the bigger car, but in the second set of equations it represents the relative speed of the moving frame.
 
  • #3
You're right. I ended up solving the problem. Thanks anyways.

Consider this problem solved.
 

Related to Show momentum is conserved in two different frames (relativity)

What is momentum and why is it important in relativity?

Momentum is a measure of an object's motion, calculated by multiplying its mass by its velocity. In relativity, momentum is important because it is conserved, meaning it remains constant in a closed system, regardless of the frame of reference. This helps explain the behavior of objects in different frames and is a fundamental principle in understanding the laws of motion.

What is the principle of relativity and how does it relate to momentum conservation?

The principle of relativity states that the laws of physics are the same in all inertial reference frames. This means that the momentum of an object will be conserved in all frames of reference, as long as there are no external forces acting on the system. This principle is essential in understanding the concept of momentum conservation in different frames.

How is momentum conserved in two different frames?

In a closed system, the total momentum of all objects will remain constant in any frame of reference. This means that if an object's momentum increases in one frame, another object's momentum must decrease in the same frame, and the opposite will occur in the other frame. This ensures that the total momentum of the system remains the same in both frames, regardless of their relative velocities.

Does momentum conservation hold true in both classical and relativistic mechanics?

Yes, the principle of momentum conservation holds true in both classical and relativistic mechanics. However, in relativity, momentum is not an absolute quantity, and its value can change depending on the frame of reference. This is due to the effects of time dilation and length contraction in relativistic systems.

What experiments have been conducted to demonstrate momentum conservation in different frames?

Several experiments have been conducted to demonstrate the conservation of momentum in different frames, such as the famous Michelson-Morley experiment, which showed that the speed of light remains constant in all inertial frames. Other experiments, such as the Compton scattering experiment and the particle accelerator experiments, have also confirmed the principle of momentum conservation in different frames.

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