Newton's first law

Newton's first law of motion, also known as the law of inertia, states that an object will remain at rest or move in a straight line with constant velocity unless acted upon by an external force. This law can be expressed as:

Statement of Newton's First Law: 

An object at rest will remain at rest, and an object in motion will continue to move at a constant velocity in a straight line unless acted upon by an external force.

Derivation:

To derive Newton's first law, let's consider two scenarios:

1. An object at rest.
2. An object moving with constant velocity.

For the first scenario, if an object is at rest, it means that its velocity is zero. According to Newton's first law, if there is no net external force acting on the object, it will remain at rest. Mathematically, this can be expressed as , where represents the sum of all forces acting on the object. Since there is no acceleration , the net force on the object is zero.

For the second scenario, if an object is moving with constant velocity, it means that its acceleration is zero . According to Newton's second law , if the acceleration is zero, the net force acting on the object must also be zero . Thus, if no external force is acting on an object, it will continue to move with constant velocity.

In both scenarios, the key point is that in the absence of external forces, there is no change in the object's state of motion. This principle forms the basis of Newton's first law of motion, which states that objects tend to maintain their state of motion (either at rest or in motion with constant velocity) unless acted upon by an external force.