(The question is too long so it's marked at the top of the answer space, sorry for any inconveniences)

Question:Particle P of mass m and particle Q of mass km are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision the speed of P is 5u and the speed of Q is u. Immediately after the collision the speed of each particle is halved and the direction of motion of each particle is reversed.Find the value of k
Answer: (will be accompanied with sketches on the whiteboard)This is a classic question about collision between 2 particles, in which the two underlying principles are conservation of energy and momentum. Setting up the problem we draw P and Q marking their speed and mass of m and km, before and after the collision. before:(P travelling at 5u to the right and Q travelling at u to the left) We know each particle's speed and direction was halved and reversed after the collision so we can calculate them respectively to be 5u/2 to the left for P and u/2 to the right for Q. Now we need to think about which rule to apply for this situation. We were given the speeds and masses of the two particles which enables us to work out the kinetic energy or the momentum of the particles. It is important to note that kinetic energy in a collision are not always conserved, as it can be converted to other forms of energy, while keeping the total energy constant and satisfying the conservation of energy. In fact, KE is only conserved for elastic collisions. Thus, we should use the conservation of momentum to solve this problem.Taking towards the right as the positive direction we calculate the total momentum before and after the collision. momentum of a particle is given by their mass x speed so before the collision the total momentum is 5mu-kmu which simplifies to (5-k)mu.(notice negative sign of kmu(momentum of Q) due to it travelling towards the left) After the collision we calculate the total momentum which gives kmu/2 - 5mu/2 -> (k/2 - 5/2)mu. Applying conservation of momentum and cancelling out the common factor of mu gives 5-k = (k-5)/2 which we solve to find k = 5