Inertial Frame R: Persisting Relationships of Particles' Positions & Velocities

[imy786]

Suppose that we choose an inertial frame R in which the particles’ positions and velocities are related by

A1= - m2 (A2) / m1


V1 = - m2(V2) / m1

at time t = 0. Show that these relationships persist at all subsequent times.

 

[robphy]

sounds like homework

 

[Entropia]

These relationships between the positions and velocities of particles in an inertial frame R are known as the equations of motion, and they are fundamental in understanding the behavior of particles in a given system. The equations state that the acceleration of a particle is directly proportional to the net force acting on it, and inversely proportional to its mass. In this case, we have two particles with masses m1 and m2, and their positions and velocities are related by the equations A1= - m2 (A2) / m1 and V1 = - m2(V2) / m1 at time t = 0.

To show that these relationships persist at all subsequent times, we can use the laws of motion and the principle of superposition. According to the first law of motion, an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. In this case, the particles are initially at rest, so they will remain at rest in the absence of any external forces.

At time t = 0, if we apply a force F on the first particle, according to the second law of motion, it will experience an acceleration A1 = F/m1. This acceleration will also be equal to -m2(A2)/m1, as stated in the given equations. This means that the force applied on the first particle is equal in magnitude and opposite in direction to the force exerted by the second particle, as expected from Newton's third law of motion.

Now, if we consider the second particle, it will experience an acceleration A2 = -F/m2, which is also equal to -m1(A1)/m2. This shows that the force exerted by the first particle on the second particle is also equal in magnitude and opposite in direction to the force applied on the first particle.

Applying the principle of superposition, we can extend this reasoning to any subsequent time. If we apply a force on either particle, the same relationships between the forces, accelerations, and masses will hold, and the equations A1= - m2 (A2) / m1 and V1 = - m2(V2) / m1 will continue to be valid.

In conclusion, the relationships between the positions and velocities of particles in an inertial frame R will persist at all subsequent times, as long as the particles remain in an inertial frame and there are no external forces acting on them. This shows the