Final answer:
Final velocities after an elastic collision can be found using conservation principles of momentum and kinetic energy. The ratio of final to initial kinetic energy for a mass can be computed, and the change in momentum for each mass and the system can be determined, with the system's total change in momentum being zero.
Step-by-step explanation:
When two masses undergo a perfectly elastic head-on collision, their final velocities can be calculated using the principles of conservation of momentum and conservation of kinetic energy. For mass 'm' with an initial velocity of 50 m/s in the +ve x direction and mass '2m' with an initial velocity of 40 m/s in the -ve x direction, the system's total momentum and kinetic energy before the collision must equal that after the collision. Since this is a perfectly elastic collision, no kinetic energy is lost.
To find the final velocities, we need to set up two equations: one for the conservation of momentum (momentum before collision equals momentum after collision) and one for the conservation of kinetic energy (kinetic energy before collision equals kinetic energy after collision). Solving this system of equations will give us the final velocities of both masses.
The change in momentum of each mass can be calculated by taking the difference between the final and initial momentums. The change in momentum of the system, however, should be zero, because momentum is conserved in the absence of external forces.
The ratio of the final kinetic energy (K.E.) to the initial kinetic energy for mass 'm' is calculated by taking the kinetic energy after the collision and dividing it by the kinetic energy before the collision, using the formula for kinetic energy, which is (1/2)mv2.