A 2kg mass moving with a velocity of 7 m/s collides elastically with a 4 kg mass moving in the opposite direction at 4 m/s. The 2 kg mass reverses direction after the collision and has a new velocity of 3 m/s. What is the new velocity of the 4 kg mass?

My first approach was with conservation of kinetic energy.
Then I got the correct answer using the conservation of momentum formula.

However, why do I not get this same answer using conservation of kinetic energy? Both kinetic energy and momentum are conserved in an elastic collision...I'm accounting for direction changes in the conservation of kinetic energy formula as well...

And since apparently I cannot use conservation of kinetic energy for this, when DO I use conservation of kinetic energy?

The problem is fake. It is not an elastic collision. You can prove that by not assigning a velociy for after the collision, then solving for it using both equations.

conservation of momentum holds for elastic AND inelastic equations.

Anyway, ALWAYS use momentum first. You can not count on the energy being the same before and after unless the collision is elastic. I suspect that the claim of elasticity was a typo.

In an elastic collision, both momentum and kinetic energy are conserved. However, it's important to note that the conservation of kinetic energy only applies when the collision is one-dimensional and the objects involved have no rotational motion.

Let's analyze why conservation of kinetic energy doesn't give the correct answer in this particular scenario.

Initially, the 2 kg mass is moving with a velocity of 7 m/s, and the 4 kg mass is moving in the opposite direction with a velocity of 4 m/s. After the collision, the 2 kg mass reverses direction and has a new velocity of 3 m/s. We need to find the new velocity of the 4 kg mass.

Using the conservation of kinetic energy:

Initially, the total kinetic energy is given by the sum of the kinetic energies of the two masses:
KE_initial = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
= (1/2) * 2 kg * (7 m/s)^2 + (1/2) * 4 kg * (4 m/s)^2
= 98 J + 32 J
= 130 J

After the collision, the total kinetic energy is given by:
KE_final = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2
= (1/2) * 2 kg * (3 m/s)^2 + (1/2) * 4 kg * v2'^2
= 9 J + 2 kg * v2'^2

Since we know that the kinetic energy is conserved in an elastic collision, we can equate the initial and final kinetic energies:
KE_initial = KE_final
130 J = 9 J + 2 kg * v2'^2

Solving this equation, we find that 2 kg * v2'^2 = 121 J, which results in v2'^2 = 60.5 J/kg.

As you can see, we end up with an equation that relates the square of the velocity of the 4 kg mass after the collision (v2'^2) to a value in Joules (J/kg), which is not directly related to velocity.

Therefore, when trying to find the new velocity of the 4 kg mass using the conservation of kinetic energy, we encounter a limitation. This limitation occurs because kinetic energy is a scalar quantity (no direction) and does not account for changes in velocity direction.

In summary, you should use the conservation of kinetic energy when you want to analyze a collision where the direction of the masses' motion remains unchanged. For situations involving changes in direction like in this case, conservation of momentum is the appropriate principle to use.