Mechanics

Step 1: Understanding the Concept:
The question contains a contradiction. A "perfectly elastic collision" is one where kinetic energy is conserved. A collision where "particles stick together" is a "perfectly inelastic collision", where kinetic energy is not conserved, but momentum is. Since the question asks for the final velocity under the condition that the particles stick together, we must treat it as a perfectly inelastic collision and disregard the "perfectly elastic" part. The governing principle for all collisions in an isolated system is the conservation of linear momentum.
Step 2: Key Formula or Approach:
The principle of conservation of linear momentum states that the total momentum before the collision is equal to the total momentum after the collision.
where is the final common velocity of the combined mass.
Step 3: Detailed Explanation:
Let the initial velocities of the two particles be and . The total momentum before the collision is .
After the collision, the particles stick together, so they move as a single object with mass and a final common velocity .
The total momentum after the collision is .
By conserving momentum:
Solving for the final velocity :
Step 4: Final Answer:
The velocity after the collision when the particles stick together is given by the formula for the velocity of the center of mass, which matches option (B).

Step 1: Understanding the Concept:
This problem involves relative motion and requires the use of Galilean velocity transformation. We are given the position of a particle in one inertial frame (S) and the velocity of another inertial frame (S') relative to S. We need to find the velocity of the particle as observed from frame S'.
Step 2: Key Formula or Approach:
The Galilean transformation for velocities relates the velocity of a particle in frame S ( ), the velocity of the particle in frame S' ( ), and the velocity of frame S' relative to S ( ):
To find the velocity of the particle in frame S', we rearrange this formula:
Step 3: Detailed Explanation:
1. Find the velocity of the particle in frame S ( ):
The position vector of the particle in frame S is given by:
To find the velocity, we differentiate the position vector with respect to time :
2. Identify the velocity of frame S' relative to S ( ):
This is given in the problem:
3. Calculate the velocity of the particle in frame S' ( ):
Using the transformation formula:
Step 4: Final Answer:
The velocity of the particle in frame S' is m/s. This corresponds to option (A).