To solve this problem, we can apply the principles of conservation of momentum and conservation of kinetic energy for an elastic collision.
Step 1: Find the initial total momentum (Pi) of the system.
The momentum of an object is given by the equation:
Momentum = mass * velocity
For the first block:
Mass1 = 0.104 kg
Velocity1 = 4.70 m/s
Momentum1 = Mass1 * Velocity1
For the second block:
Mass2 = 0.208 kg
Velocity2 = -6.50 m/s (since the second block is moving in the opposite direction)
Momentum2 = Mass2 * Velocity2
The initial total momentum (Pi) of the system is the sum of the individual momenta of the blocks:
Pi = Momentum1 + Momentum2
Step 2: Find the initial total kinetic energy (Ki) of the system.
The kinetic energy of an object is given by the equation:
Kinetic energy = (1/2) * mass * (velocity^2)
For the first block:
Kinetic energy1 = (1/2) * Mass1 * (Velocity1^2)
For the second block:
Kinetic energy2 = (1/2) * Mass2 * (Velocity2^2)
The initial total kinetic energy (Ki) of the system is the sum of the individual kinetic energies of the blocks:
Ki = Kinetic energy1 + Kinetic energy2
Step 3: Apply the conservation of momentum and conservation of kinetic energy equations for an elastic collision.
When two objects collide elastically, the total momentum and total kinetic energy before the collision are equal to the total momentum and total kinetic energy after the collision.
Conservation of momentum:
Pi = Pf, where Pf is the final total momentum of the system.
Conservation of kinetic energy:
Ki = Kf, where Kf is the final total kinetic energy of the system.
Step 4: Solve the equations.
Based on the conservation of momentum:
Pi = Pf
Momentum1 + Momentum2 = Momentum1' + Momentum2', where the primes (') denote the final velocities.
Based on the conservation of kinetic energy:
Ki = Kf
Kinetic energy1 + Kinetic energy2 = Kinetic energy1' + Kinetic energy2', where the primes (') denote the final kinetic energies.
Using the formulas for momentum and kinetic energy, substitute the given values into these equations:
(Mass1 * Velocity1) + (Mass2 * Velocity2) = (Mass1 * Velocity1') + (Mass2 * Velocity2')
(1/2) * Mass1 * (Velocity1^2) + (1/2) * Mass2 * (Velocity2^2) = (1/2) * Mass1 * (Velocity1'^2) + (1/2) * Mass2 * (Velocity2'^2)
Now, we need to solve these equations to find the final velocities of the blocks. The calculations involve quadratic equations.