Ball A, moving with an initial speed of 3 m/s, has a head-on collision with another ball B at rest, which is twice as massive as A . What are the final velocities of A and B if the collision is elastic?

To find the final velocities of the balls after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.

1. Conservation of momentum:
The total momentum before the collision is equal to the total momentum after the collision. The momentum of an object is given by the product of its mass and velocity: momentum = mass × velocity.

Before the collision:
Ball A's initial momentum = (mass of A × velocity of A) = (mass of A × 3 m/s)
Ball B's initial momentum = (mass of B × velocity of B) = (2 × mass of A × 0 m/s) (since B is at rest)

After the collision:
Ball A's final momentum = (mass of A × velocity of A')
Ball B's final momentum = (mass of B × velocity of B')

Using the principle of conservation of momentum, we equate the total initial momentum to the total final momentum:
(mass of A × 3 m/s) = (mass of A × velocity of A') + (2 × mass of A × velocity of B')

2. Conservation of kinetic energy:
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. The kinetic energy of an object is given by the formula: kinetic energy = (1/2) × mass × velocity².

Before the collision:
Ball A's initial kinetic energy = (1/2) × (mass of A) × (3 m/s)²
Ball B's initial kinetic energy = 0 (since B is at rest)

After the collision:
Ball A's final kinetic energy = (1/2) × (mass of A) × (velocity of A')²
Ball B's final kinetic energy = (1/2) × (mass of B) × (velocity of B')²

Using the principle of conservation of kinetic energy, we equate the total initial kinetic energy to the total final kinetic energy:
(1/2) × (mass of A) × (3 m/s)² = (1/2) × (mass of A) × (velocity of A')² + (1/2) × (mass of B) × (velocity of B')²

Solving these two equations simultaneously will allow us to find the final velocities of ball A (velocity of A') and ball B (velocity of B') after the elastic collision.

In an elastic collision, both momentum and kinetic energy are conserved.

Conservation of momentum

m*3 = m*vA + 2m*vB
or
3 = vA + 2*VB
where vA is the speed of A, and vB is the speed of B

Conservation of kinetic energy:

1/2*m*3^2 = 1/2*m*vA^2 + 1/2*2*m*vB^2
9 = vA^2 + 2*vB^2

You have two equations with two unknowns:

3 = vA + 2*VB
9 = vA^2 + 2*vB^2

Use algebra to solve for vA and vB