Principles in the formation of the Equation of Motion of a Particle :-
For Motion in Straight Line ;
Let the mass of the body (particle) be m and let the distance of the particle measured from a suitable origin be x at the time t .
Then acceleration is (d/dt)(dx/dt)
therefore by Second Law of Motion
m[(d/dt)(dx/dt)] = Forces in the direction of x increasing
The forces usually are ;-
1)--- g vertically downwards in a gravitational field like earth .
2)--- tension in a string
3)--- reactions or stress at point where the point may be in contact with other particles .
4)--- forces of attraction
5)--- forces of resistance to motion by any means may be by atmosphere , friction, winds or by any other means .
In taking the forces we must fix the sign properly if the forces acts along the line .
If a force F acts at an angle A to the straight line , then the resolved part of the force along the line is FcosA . If g acts on on a particle and if the particle is moving along the horizontal line , then g , being vertical , has no resolved part horizontally and g does not effects the Motion .
Also note that velocity, v=dx/dt , and acceleration
(d/dt)(dx/dt) = dv/dt = (dx/dt)(dv/dx) = v(dv/dx)
For Motions in two dimension;
Let two perpendicular axes are X and Y at the plane of motion . The forces are resolved in these two directions and respectively
m(d/dt)(dx/dt) and m(d/dt)(dy/dt)