A small particle of mass m travelling with speed v collide elastically with a spherical body of equal mass and radius r,initially at rest.The centre of spherical body at a distance rho(rho is smaller than r) away from direction of motion of particle. The velocity of the two objects after collision,would be

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To solve this problem, we need to analyze the elastic collision between a small particle and a spherical body. Since both objects have equal mass and the collision is elastic, we can apply the principles of conservation of momentum and kinetic energy. Let's break this down step by step.

Understanding the Setup

We have two objects: a small particle of mass m moving with speed v, and a spherical body of equal mass m that is initially at rest. The center of the spherical body is located at a distance ρ from the path of the particle, where ρ is less than the radius r of the spherical body. This means that the particle will collide with the spherical body at an angle.

Key Concepts

• Elastic Collision: Both momentum and kinetic energy are conserved.
• Momentum Conservation: The total momentum before the collision equals the total momentum after the collision.
• Kinetic Energy Conservation: The total kinetic energy before the collision equals the total kinetic energy after the collision.

Applying Conservation Laws

Let’s denote the initial velocity of the particle as v and the initial velocity of the spherical body as 0 (since it is at rest). After the collision, we will denote the velocities of the particle and the spherical body as v_p and v_s, respectively.

Momentum Conservation Equation

The momentum before the collision can be expressed as:

m * v + m * 0 = m * v_p + m * v_s

This simplifies to:

v = v_p + v_s

Kinetic Energy Conservation Equation

The kinetic energy before the collision is:

0.5 * m * v² + 0.5 * m * 0 = 0.5 * m * v_p² + 0.5 * m * v_s²

This simplifies to:

0.5 * v² = 0.5 * v_p² + 0.5 * v_s²

We can further simplify this to:

v² = v_p² + v_s²

Solving the Equations

Now we have two equations:

• 1. v = v_p + v_s
• 2. v² = v_p² + v_s²

From the first equation, we can express v_p in terms of v_s:

v_p = v - v_s

Substituting this into the second equation gives:

v² = (v - v_s)² + v_s²

Expanding the left side:

v² = v² - 2v * v_s + v_s² + v_s²

This simplifies to:

0 = -2v * v_s + 2v_s²

Factoring out 2v_s gives:

0 = 2v_s(v_s - v)

Finding the Velocities

This equation yields two solutions:

• v_s = 0 (the spherical body does not move, which is not the case here)
• v_s = v (the spherical body moves with the same speed as the particle)

Now substituting back to find v_p:

v_p = v - v = 0

Final Results

After the elastic collision, the spherical body moves with speed v, and the small particle comes to rest. Thus, the velocities after the collision are:

• Velocity of the small particle (v_p): 0
• Velocity of the spherical body (v_s): v

This outcome illustrates how momentum and energy conservation principles govern the behavior of colliding objects, even when they are not aligned perfectly in their paths. If you have any further questions or need clarification on any part of this explanation, feel free to ask!