Why is momentum conservation always true in a collision?

[Np14]

Homework Statement

Two blocks of mass 2M and M head toward each other sliding over a frictionless surface sliding with speeds 4v and 6v respectively. After the collision the 2m mass is at rest, and the mass M has a velocity of 2V to the right. Was the collision elastic or inelastic?

• a. Inelastic since kinetic energy wasn't conserved
• b. Inelastic since momentum wasn't conserved
• c. Elastic since kinetic energy was conserved
• d. Elastic since momentum was conserved

Homework Equations

K = 1/2mv², Δp = mv - mv_o

The Attempt at a Solution

I am sure this is an inelastic collision because one of the objects is at rest after impact so that eliminates choices c and d. But my confusion stems from the belief that kinetic energy conservation and (linear) momentum conservation go hand in hand. Can someone please explain why this is not the case?

 

[Np14]

Is momentum always conserved in linear collisions?

 

[PeroK]

Is momentum always conserved in linear collisions?

Momentum is always conserved. Kinetic Energy is not necessarily conserved.

I am sure this is an inelastic collision because one of the objects is at rest after impact so that eliminates choices c and d. But my confusion stems from the belief that kinetic energy conservation and (linear) momentum conservation go hand in hand. Can someone please explain why this is not the case?

I don't agree that you can eliminate c) or d) on that basis.

 

[Doc Al]

I am sure this is an inelastic collision because one of the objects is at rest after impact so that eliminates choices c and d.

To add to what @PeroK said: Why don't you just calculate the KE before and after the collision and compare?

But my confusion stems from the belief that kinetic energy conservation and (linear) momentum conservation go hand in hand. Can someone please explain why this is not the case?

For one thing, momentum is a vector quantity, but kinetic energy is a scalar. Example: Two objects heading toward each other (same mass, same speed, moving in opposite directions) and about to collide and stick together.

Before the collision: Momentum = zero. KE ≠ zero.
After the collision: Momentum still = zero. KE = zero.

 

[kuruman]

But my confusion stems from the belief that kinetic energy conservation and (linear) momentum conservation go hand in hand. Can someone please explain why this is not the case?

Suppose two identical blocks of mass $m$ have equal speeds $v$ and are moving in opposite directions. They collide, stick together and are at rest after the collision.
The kinetic energy before the collision is $2 \times\frac{1}{2}mv^2$ and after the collision it is zero. Therefore kinetic energy is not conserved.
The linear momentum before the collision is $mv+(-mv)=0$ and after the collision it is still zero. Therefore momentum is conserved.
Clearly the two do not go hand in hand. As @PeroK already noted, momentum is always conserved throughout a collision. That's because it is a consequence of Newton's 3rd law that always holds. Energy is not necessarily conserved and that's because there might be dissipative forces like friction that diverts some of the energy into the generation of heat. If nothing else, if you hear a collision, some of the initial kinetic energy goes into the production of sound waves ...

For future reference: if you deal with a collision problem, you should always conserve momentum. Do not assume that energy is conserved unless there is specific language in the problem indicating that.