A ball A collides head-on with another ballB of equal mass. Assuming that ball B is atrest and their collision is elastic, determinethe final velocities of these balls

To determine the final velocities of the balls after the head-on collision, we can use the principles of conservation of momentum and kinetic energy.

Step 1: Conservation of momentum Since the collision is elastic and ball B is at rest, 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.

Let the initial velocity of ball A be vA and the initial velocity of ball B be vB. Since ball B is at rest, its initial velocity vB is 0.

The total initial momentum is given by: Initial momentum = mass of ball A * initial velocity of ball A + mass of ball B * initial velocity of ball B Initial momentum = mass of ball A * vA + mass of ball B * 0 Initial momentum = mass of ball A * vA

Step 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 half the product of its mass and the square of its velocity.

The total initial kinetic energy is given by: Initial kinetic energy = (1/2) * mass of ball A * (initial velocity of ball A)^2 + (1/2) * mass of ball B * (initial velocity of ball B)^2 Initial kinetic energy = (1/2) * mass of ball A * vA^2 + (1/2) * mass of ball B * 0^2 Initial kinetic energy = (1/2) * mass of ball A * vA^2

Step 3: Applying conservation of momentum and kinetic energy Since the total momentum and total kinetic energy are conserved, we can equate the initial and final values.

From conservation of momentum: mass of ball A * vA = mass of ball A * final velocity of ball A + mass of ball B * final velocity of ball B

From conservation of kinetic energy: (1/2) * mass of ball A * vA^2 = (1/2) * mass of ball A * (final velocity of ball A)^2 + (1/2) * mass of ball B * (final velocity of ball B)^2

Step 4: Solving the equations Since the masses of ball A and ball B are equal, we can simplify the equations:

mass of ball A * vA = mass of ball A * final velocity of ball A + mass of ball A * final velocity of ball B vA = final velocity of ball A + final velocity of ball B

(1/2) * mass of ball A * vA^2 = (1/2) * mass of ball A * (final velocity of ball A)^2 + (1/2) * mass of ball A * (final velocity of ball B)^2 vA^2 = (final velocity of ball A)^2 + (final velocity of ball B)^2

Step 5: Solving for the final velocities Substituting the first equation into the second equation, we get:

vA^2 = (vA + final velocity of ball B)^2

Expanding and simplifying the equation, we get:

vA^2 = vA^2 + 2 * vA * final velocity of ball B + (final velocity of ball B)^2

Simplifying further, we get:

0 = 2 * vA * final velocity of ball B + (final velocity of ball B)^2

This is a quadratic equation in terms of final velocity of ball B. Solving this equation will give us the final velocities of the balls.