Introduction:
In this scenario, we have a moving particle of mass m that collides perfectly inelastically with another stationary particle of mass M=2m. We are given that the incident particle deflects by 90° after the collision. Our task is to determine the angle made by the heavy mass (M) with the initial direction of the moving particle (m).

Analysis:
To solve this problem, we can consider the conservation of momentum and the conservation of kinetic energy during the collision.

Conservation of Momentum:
Before the collision, the momentum of the system is given by the sum of the momenta of the two particles:
m₁u₁ + M₂u₂ = (m + M)V

After the collision, the two particles move together as one, so their final velocity is the same:
(m + M)V = (m + M)vf

Conservation of Kinetic Energy:
Before the collision, the kinetic energy of the system is given by the sum of the kinetic energies of the two particles:
(1/2)m₁u₁² + (1/2)M₂u₂² = (1/2)(m + M)V²

After the collision, the two particles stick together and move with the same final velocity:
(1/2)(m + M)V² = (1/2)(m + M)vf²

Solution:
Since the incident particle deflects by 90° after the collision, it means that the final velocity of the system is perpendicular to the initial direction of the moving particle. Therefore, the final velocity vf is horizontal.

Using the conservation of momentum equations, we can simplify them by substituting the final velocity vf:

m₁u₁ + M₂u₂ = (m + M)V
(m + M)V = (m + M)vf

Now, we can eliminate V and vf by dividing the two equations:

m₁u₁ + M₂u₂ = (m + M)vf / (m + M)

Simplifying further:

m₁u₁ + M₂u₂ = vf

We can see that the final velocity vf is equal to the initial velocity of the moving particle m₁u₁ plus the initial velocity of the stationary particle M₂u₂.

Now, to find the angle made by the heavy mass (M) with the initial direction of the moving particle (m), we can use trigonometry. Let's assume that the angle we are looking for is θ.

Using the trigonometric relationship:

tan(θ) = (M₂u₂) / (m₁u₁)

Substituting the given values, where M = 2m:

tan(θ) = (2m)(0) / (mu₁)

tan(θ) = 0

Therefore, the angle made by the heavy mass (M) with the initial direction of the moving particle (m) is 0°.

Summary:
In summary, when a moving particle of mass m collides perfectly inelastically with a stationary particle of mass M=2m and the incident particle deflects by 90°, the heavy mass makes an angle of 0° with the initial direction of the moving particle. This means that the heavy mass does not change its direction and remains aligned with the initial direction of the moving particle

if they collide perfectly inelastically then they will move together.