Given data:
- Mass of the particle (m) = 2.5 kg
- Initial velocity of the particle (v) = 3 m/s
- Magnitude of the force acting on the particle (F) = 67.5 N

Problem statement:
Find the minimum time after which the particle has the same velocity as the initial velocity.

Solution:
To find the minimum time required for the particle to have the same velocity as the initial velocity, we need to analyze the given situation.

Understanding the problem:
- The force acting on the particle is always perpendicular to its velocity. This means that the force is exerted in a direction perpendicular to the direction of motion.
- In the absence of external forces, the velocity of an object remains constant in magnitude and direction.
- Since the force is always perpendicular to the velocity, it does not change the magnitude of the velocity, but it can change the direction of the velocity.

Analysis:
- The force acting on the particle is perpendicular to its velocity. This implies that the force is providing a centripetal acceleration to the particle.
- The centripetal force required to keep a particle moving in a circular path is given by the formula Fc = (mv^2) / r, where Fc is the centripetal force, m is the mass of the particle, v is the velocity of the particle, and r is the radius of the circular path.
- In this case, the force acting on the particle is perpendicular to its velocity, which means that the particle is moving in a circular path of radius r.
- Since the force acting on the particle is always perpendicular to its velocity, the direction of the force is always changing. This implies that the particle is moving in a circular path with a continuously changing radius.
- Since the force is always perpendicular to the velocity, the magnitude of the velocity remains constant.

Finding the minimum time:
- The magnitude of the force acting on the particle is given as 67.5 N.
- The magnitude of the force required to keep the particle moving in a circular path is given as Fc = (mv^2) / r.
- Setting these two forces equal, we get 67.5 = (2.5 * 3^2) / r.
- Solving this equation for r, we get r = (2.5 * 3^2) / 67.5.
- The minimum time required for the particle to have the same velocity as the initial velocity is equal to the time taken for the particle to complete one revolution in this circular path.
- The time period of a circular motion is given by T = 2πr / v, where T is the time period, r is the radius of the circular path, and v is the velocity of the particle.
- Substituting the values, we get T = (2π * [(2.5 * 3^2) / 67.5]) / 3.
- Simplifying this expression, we get T = (2π * 9) / 67.5.
- Therefore, the minimum time after which the particle has the same velocity as the initial velocity is (2π * 9) / 67.5 seconds.