# Conservation of Momentum in Different Frames of Reference

• quantised
In summary, the conversation discusses the calculation of the final speed of a bullet fired from a gun from both a stationary observer's frame of reference and the gun's frame of reference. The problem with using the gun's frame of reference is that it is not an inertial frame, meaning that the laws of physics, such as conservation of momentum, do not apply without modification. Therefore, the final speed calculated in this frame will be incorrect.
quantised
Hello All,

The following may be a simple problem. But, your thoughts will be very much appreciated.

## Homework Statement

Let's use a gun with mass m1 and a bullet m2. The bullet is fired in the positive direction with speed v2, and the gun recoils in the negative direction with speed v1.

Because the momentum is conserved, m1v1 = m2v2.

Hence, v2 can be calculated if m1, m2 and v1 are all known. In other words, v2 = m1v1 / m2. This calculation is from a stationary observer's frame of reference, outside the gun/bullet system.

## Homework Equations

Now, I want to transform the same problem to the gun's frame of reference. (Hypothetically, it's an ant or a small person sitting at the muzzle observing the event!)

In this new frame of reference, the bullet moves at the following final speed: vb, final = v2 - v1. The initial speed vinitial is zero, because the gun and the bullet are stationary. Therefore:

m2 (v2 - v1) = (m1 + m2) vinitial = 0

This means that v1 = v2.

## The Attempt at a Solution

Clearly, the answer from the first frame should be equal to the answer in the second frame.

Can someone please indicate to me where my working is incorrect?

Quantised

Last edited:
The problem is that your chosen frame of reference is not an inertial frame -- it will experience an acceleration while the gun is being fired, which means there will be some transient pseudo forces operating on all objects measured from that frame. Physical laws (like conservation of momentum) do not apply without modification to non-inertial frames of reference.

## 1. What is conservation of momentum in different frames of reference?

The law of conservation of momentum states that the total momentum of a closed system remains constant in both magnitude and direction, regardless of any external forces acting on the system. This means that in different frames of reference, the total momentum of the system will remain the same.

## 2. How is momentum conserved in different frames of reference?

In different frames of reference, momentum is conserved through the principle of relativity. This means that the laws of physics, including the conservation of momentum, hold true in all inertial frames of reference. This allows us to use different reference frames to analyze the same system and still come to the same conclusion about the conservation of momentum.

## 3. What is an inertial frame of reference?

An inertial frame of reference is a frame of reference in which a body at rest remains at rest and a body in motion continues to move at a constant velocity in a straight line, unless acted upon by an external force. In other words, there are no unbalanced forces acting on the system in an inertial frame of reference.

## 4. Can conservation of momentum be violated in different frames of reference?

No, conservation of momentum is a fundamental law of physics that holds true in all inertial frames of reference. However, in non-inertial frames of reference, such as a accelerating or rotating frame, the conservation of momentum may appear to be violated due to the presence of fictitious forces.

## 5. How does the conservation of momentum apply to collisions in different frames of reference?

The conservation of momentum applies to collisions in different frames of reference in the same way it does in a fixed frame of reference. In a collision, the total momentum of the system before and after the collision must be equal, regardless of the frame of reference used to analyze it. This allows us to accurately predict the outcomes of collisions in different frames of reference using the principles of conservation of momentum.

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