Conservation of linear momentum in RH relations

In summary, the conservation of linear momentum in RH (right-hand) relations refers to the principle that in a closed system, the total linear momentum remains constant when no external forces act upon it. This concept is pivotal in understanding various physical processes and interactions, as it allows for the prediction of outcomes in collisions and other dynamics. The RH relations specifically pertain to the directional aspects of momentum conservation, emphasizing the importance of vector quantities in analyzing motion and ensuring that momentum is conserved in all frames of reference.
  • #1
GeologistInDisguise
6
1
TL;DR Summary
How is the pictured equation a conservation of momentum equation and what does pressure have to do with it?
I am trying to follow a derivation of the Rankine-Hugoniot equations in a paper by Peter Krehl titled:

The classical Rankine-Hugoniot jump conditions, an important cornerstone of modern shock wave physics: ideal assumptions vs. reality
1699572663390.png

This paper talks about the RH equations which relate kinematic properties to thermodynamic ones when a shock is transiting a material. See photo above for an illustration. In section 2.2.2 when discussing this relation in terms of Lagrangian coordinates, equation 4a is introduced and described as conservation of linear momentum:

1699571651118.png


Where P is pressure, rho is density, u is particle velocity, 0 indicates downstream or preshock and 1 indicates upstream or post shock.

While the units do make sense, how is this conservation of momentum? if momentum is m*v, I suppose you could divide by volume to replace mass with density, but then why is velocity squared? And why is pressure being added?

I understand pressure is force/area and also that force is the change in momentum with time. I am guessing this has something to do with it but I am not getting anything that would give me a squared velocity. Obviously if I integrate my mass normalized momentum equation with respect to time, that would give me a velocity squared. But why would I do that?
 
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  • #2
ΔP is a force which changes momentum
ρu2 is the amount of momentum that is flowing into/out of the box (think m=ρuAΔt)
the AΔt is also attached to the pressures, so it cancels out
 
  • #3
Frabjous said:
ΔP is a force which changes momentum
ρu2 is the amount of momentum that is flowing into/out of the box (think m=ρuAΔt)
the AΔt is also attached to the pressures, so it cancels out
I think that makes sense, but why then is velocity squared?
 
  • #4
GeologistInDisguise said:
I think that makes sense, but why then is velocity squared?
The “mass” has a velocity in it. You then have to multiply it by velocity to get momentum.
 
  • #5
Frabjous said:
The “mass” has a velocity in it. You then have to multiply it by velocity to get momentum.
Oh, in your previous comment was m mass or momentum? I assumed you meant it was momentum since we were using P for pressure already but I think this makes more sense if you did actually mean mass.
 
  • #6
Particle KE = ½·m·v² ≈ absolute temperature.
(speed of sound)² ≈ Tabs
The speed of sound is a direct function of √Tabs only.
Temperature is an indirect function of pressure.
 
  • #7
Baluncore said:
Particle KE = ½·m·v² ≈ absolute temperature.
(speed of sound)² ≈ Tabs
The speed of sound is a direct function of √Tabs only.
Temperature is an indirect function of pressure.
This is an interesting point but I am not seeing how it is relevant to the question posted about conservation of momentum?
 
  • #8
GeologistInDisguise said:
Oh, in your previous comment was m mass or momentum? I assumed you meant it was momentum since we were using P for pressure already but I think this makes more sense if you did actually mean mass.
I meant mass.
The equation holds in the reference frame where the shock velocity is zero. So there is a discrepancy between the picture and the equation. I believe the picture should have Us=0.
There are several versions of the equation. Which one is useful depends on the application.
 
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FAQ: Conservation of linear momentum in RH relations

What is the principle of conservation of linear momentum?

The principle of conservation of linear momentum states that the total linear momentum of a closed system remains constant if no external forces are acting on it. This means that the momentum before an event (such as a collision) is equal to the momentum after the event, provided that the system is isolated from external influences.

How does conservation of linear momentum apply to RH relations?

In the context of RH relations, which often refer to right-hand rules in physics, conservation of linear momentum can be used to analyze interactions between particles or objects. For example, when applying the right-hand rule to determine the direction of momentum vectors, one can predict the resulting momentum after collisions or interactions, ensuring that the total momentum remains conserved.

Can you give an example of conservation of linear momentum in a collision?

Sure! Consider a two-object collision where object A with mass m1 is moving towards object B with mass m2 at rest. Before the collision, the total momentum is m1 * v1 (where v1 is the velocity of A). After the collision, if they stick together, the total momentum can be expressed as (m1 + m2) * v_final. According to the conservation of linear momentum, m1 * v1 = (m1 + m2) * v_final, allowing us to solve for the final velocity.

What role do external forces play in the conservation of linear momentum?

External forces can disrupt the conservation of linear momentum. If an external force acts on a system, it can change the total momentum of that system. For momentum to be conserved, the system must be isolated from external influences. Therefore, when analyzing momentum in a system, it's crucial to identify and account for any external forces that may be present.

How can the conservation of linear momentum be experimentally verified?

Conservation of linear momentum can be experimentally verified through controlled experiments, such as collision experiments using air tracks or dynamics carts. By measuring the velocities and masses of colliding objects before and after the collision, one can calculate the momentum and verify that the total momentum before the collision equals the total momentum after the collision, confirming the conservation principle.

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