A particle of mass m and initial velocity v collides elastically with a particle of

mass M at rest. After the collision, the particle of mass m moves off at an angle of θ with respect to its initial direction. How can we calculate the final velocity and direction of each particle after the collision?

To calculate the final velocity and direction of each particle after an elastic collision, we can use the principles of conservation of momentum and conservation of kinetic energy.

Step 1: Conservation of Momentum:
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. In this case, we can write the equation as:

m * v = m1 * v1 + m2 * v2

where,
m = mass of the particle initially in motion (with mass m)
v = initial velocity of the particle with mass m
m1 = mass of the particle after the collision (with mass m)
v1 = final velocity of the particle with mass m
m2 = mass of the particle initially at rest (with mass M)
v2 = final velocity of the particle with mass M

Step 2: Conservation of Kinetic Energy:
Elastic collisions preserve kinetic energy. We can write the equation as:

(1/2) * m * (v^2) = (1/2) * m1 * (v1^2) + (1/2) * m2 * (v2^2)

Step 3: Solving the equations:
We have two equations from the conservation of momentum and conservation of kinetic energy. By rearranging and solving these equations simultaneously, we can find the final velocities v1 and v2 of each particle.

Simplifying the equations, we get:

v1 = (v * (m - M) + 2 * M * v2) / (m + M)
v2 = (v * (m + M) - 2 * m * v1) / (m + M)

where, v1 is the final velocity of the particle with mass m and v2 is the final velocity of the particle with mass M.

Once you have the values of v1 and v2, you can calculate the angle θ using trigonometry. The angle θ can be determined using the equation:

θ = tan^(-1)(v1 * sin(θ) / (v1 * cos(θ) - v2))

This equation gives us the angle at which the particle with mass m moves off with respect to its initial direction.

By following these steps and solving the equations, you can calculate the final velocities and direction of each particle after the elastic collision.