Newton's Second law

Newton's second law of motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, this law is expressed as , where is the net force acting on the object, is the mass of the object, and is the acceleration produced.

This law implies that when a force is applied to an object, it will accelerate in the direction of the force. The greater the force applied, the greater the acceleration experienced by the object. Additionally, the acceleration is inversely proportional to the mass of the object; thus, heavier objects require more force to achieve the same acceleration as lighter objects.

Derivation:

To derive Newton's second law, consider an object of mass experiencing a net force , resulting in an acceleration . According to Newton's first law, the object would remain at rest or move with constant velocity if no net force were applied. However, the presence of a net force results in an acceleration according to the second law.

Mathematically, acceleration is defined as the rate of change of velocity: , where is the change in velocity and is the change in time.

Using the definition of acceleration and rearranging the terms, we can express Newton's second law as . As the time interval approaches zero, the expression becomes , which is the instantaneous rate of change of momentum with respect to time. This leads to the more general form of Newton's second law: , where is the momentum of the object.

Thus, Newton's second law provides a quantitative relationship between the net force acting on an object, its mass, and the resulting acceleration, offering a powerful framework for understanding and predicting the motion of objects.