Conservation of Linear Momentum and Covariance

Taking the derivative of x - vt with respect to time gives -v, and taking the derivative of -v with respect to t gives -a. So the velocity transformation for an observer in S would be v = v' - a. In summary, the velocity transformation for an observer in S can be derived from the Galilean transformations using the derivative with respect to time, and is given by v = v' - a.
  • #1
Cave Johnson

Homework Statement


Assume two masses m1' and m2' are moving in the positive x-direction with velocities v1' and v2' as measured by an observer in S' before a collision. After the collision, the two masses stick together and move with velocity v' in S'. Show that if an observer in S' finds momentum conserved, so does an observer in S.

Homework Equations


Galilean Transformation:
x' = x - vt
y' = y
z' = z
t' = t

Conservation of momentum in inelastic collisions:
m1v1 + m2v2 = (m1 + m2)vf

Linear momentum:
p = mv

The Attempt at a Solution


I know that this will involve the use of this part of the GT:
x' = x - vt

I am confused on how to incorporate the conservation of momentum equation(s) into this, however.

Any help would be appreciated.
 
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  • #2
How would you transform the velocities?
 
  • #3
Doc Al said:
How would you transform the velocities?

I am not sure. I don't quite understand how to use these transformations for anything other than coordinates (like measuring lengths). I try to find examples in my textbook or online but they are all very confusing or blocked by a pay wall...
 
  • #4
Cave Johnson said:
I don't quite understand how to use these transformations for anything other than coordinates (like measuring lengths).
Given the coordinate transformations, you can derive the velocity transformations by taking the derivative with respect to time. (It's easy!)
 
  • #5
Doc Al said:
Given the coordinate transformations, you can derive the velocity transformations by taking the derivative with respect to time. (It's easy!)

Wouldn't that just leave us with -v ?
 
  • #6
Cave Johnson said:
Wouldn't that just leave us with -v ?
Nope. Write the x-coordinate transform and take the derivative of each term.
 
  • #7
Doc Al said:
Nope. Write the x-coordinate transform and take the derivative of each term.

Taking the derivative of x - vt with respect to time gives -v...
d/dt x = 0
d/dt -vt = -v
 
  • #8
Cave Johnson said:
Taking the derivative of x - vt with respect to time gives -v...
d/dt x = 0
d/dt -vt = -v
Careful! The derivative of x with respect to t is not zero. It's dx/dt, which is a velocity measured in the S frame. Try it once more.
 

1. What is the conservation of linear momentum?

The conservation of linear momentum is a fundamental law in physics that states that the total momentum of a closed system remains constant over time, unless acted upon by an external force. This means that the total momentum of all objects in a system before a collision or interaction will be equal to the total momentum after the collision or interaction.

2. How does conservation of linear momentum relate to covariance?

Covariance is a mathematical concept that measures how two variables change together. In the context of conservation of linear momentum, covariance is used to show how the momenta of different objects in a system are related. Specifically, covariance can be used to show how the momentum of one object changes in response to the momentum of another object in a collision or interaction.

3. What are some real-world examples of conservation of linear momentum?

Conservation of linear momentum can be observed in many everyday situations. For example, when a billiard ball collides with another ball, the total momentum of the two balls before and after the collision will be equal. Another example is a rocket launch, where the momentum of the fuel being expelled backwards is equal to the momentum of the rocket moving forward.

4. How is the conservation of linear momentum calculated?

The conservation of linear momentum is calculated using the equation p1 + p2 = p1' + p2', where p1 and p2 represent the initial momenta of two objects and p1' and p2' represent the final momenta of those objects after a collision or interaction.

5. Is the conservation of linear momentum always true?

Yes, the conservation of linear momentum is a universal law and is always true in a closed system. However, it is important to note that this law only applies to isolated systems where there is no external force acting on the system. In reality, it is difficult to find a truly isolated system, so there may be small deviations from the law in some situations.

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