Momentum conservation inelastic collection

[UMath1]

If a car crashes with a stationary tree and comes to stop, we could say that the kinetic energy of the car was converted to heat and that the collision was inelastic. However, conservation of momentum dictates that momentum is still conserved. How would that be possible given that neither the tree nor the car possesses any velocity after the collision?

 

[A.T.]

conservation of momentum dictates that momentum is still conserved.

In an isolated system as seen from an inertial frame of reference.

 

[UMath1]

wouldn't this meet both those criteria

 

[BvU]

The logical conclusion is that momentum is not conserved ! Something exercises a force that changes the momentum... the car deforms and that requires a lot of work.

 

[A.T.]

wouldn't this meet both those criteria

What is your isolated system?

 

[UMath1]

The car and the tree

 

[BvU]

Plus the Earth underneath - or else the tree would fall over

 

[UMath1]

Right. The car, tree, and Earth underneath.

 

[UMath1]

So how is momentum conserved?

 

[BvU]

m_earth >> m_car but in principle you change the (angular) momentum of the Earth a little bit
(just as much as when you accelerate from standstill for this unhealthy experiment, only in the opposite direction)

 

[UMath1]

That doesn't quite make sense to me. If kinetic energy is transformed into thermal energy, how come the linear momentum is transformed into angular momentum of the earth. Why is it not that the macroscopic momentum of the car is transformed into the microscopic momenta of the particles that make up the car?

 

[BvU]

Ok, linear. We can safely assume the Earth is flat.

The parts of the car are welded, screwed, glued etc. together. The sum of their momenta before IS the momentum of the car.

 

[UMath1]

Right, but before the net motion of all the parts of the car is in the same direction. So we treat the car as one big particle with momentum in a given direction.

After the collision, the change is that motion of the parts of the car are in random directions and there is no net motion in any given direction, so the car has no velocity and has all its kinetic energy transformed into heat, the random motion of the microscopic particles making up the car.

My question is that why is momentum not transferred from the macroscopic motion of the car to the random microscopic motions of the particles making up the car like kinetic energy is?

 

[jbriggs444]

My question is that why is momentum not transferred from the macroscopic motion of the car to the random microscopic motions of the particles making up the car like kinetic energy is?

Because momentum is conserved. The Earth moves as a result of the collision.

In every single interaction among all the bazillions of particles that constitute the car and the earth, momentum is conserved in that interaction. Momentum is an additive property -- the momentum of the whole is the sum of the momenta of the parts. If the momentum of all of the particles is otherwise randomized, the law of conservation of momentum assures us that there is still a bias in the direction of the original bulk momentum so that total momentum is conserved. ∑mv is conserved.

Kinetic energy is also conserved in every tiny interaction. But there is no direction to kinetic energy. The law of conservation of energy only assures us that Σ½mv² is conserved. It is silent on whether the component velocities are or are not aligned.

 

[Nugatory]

My question is that why is momentum not transferred from the macroscopic motion of the car to the random microscopic motions of the particles making up the car like kinetic energy is?

The energy is independent of the direction of motion (no matter what direction a particle is moving, its kinetic energy will be $mv^2/2$) but momentum is not (a particle moving to the left will have momentum $-mv$ while a particle moving to the right will have momentum $mv$ - the net momentum of the two particles is zero).

Momentum and energy must both be conserved in the collision. Thus, after the collision all the collision fragments will collectively have the same net momentum as the original projectile. Even if the collision were to completely vaporize both objects so all that all that is left is a cloud hot gas, on average a few more particles will be moving more in one direction than the other so there will be some net momentum and the cloud as a whole will be moving. (You can see this in online videos of high-speed projectiles penetrating armor plate - a flare of fast-moving incandescently hot gas bursts out the back side, obviously carrying substantial momentum).

 

[Dale]

The car and the tree

The car and the tree do not form an isolated system. There is a very large external force exerted on the car-tree system.

Right. The car, tree, and Earth underneath.

What is the mass of the earth? How much would you expect the Earth's velocity to change? Do you think this is measurable?

 

[UMath1]

No its very insignificant owing to the large mass of the earth.

 

[UMath1]

What about if you had a car that skids to a stop. That would not exactly be a collision, but wouldn't momentum still be conserved if you considered the closed system to be the Earth and the car? Where would the momentum of the car be transferred to then?

 

[Dale]

wouldn't momentum still be conserved if you considered the closed system to be the Earth and the car?

Yes

Where would the momentum of the car be transferred to then?

Based on the answers you have received so far, what do you think? Can you justify your answer using Newtons 3rd law?