Derive the expressions for newton's law of motions.

Newton's laws of motion are three physical laws that form the foundation for classical mechanics. They describe the relationship between the motion of an object and the forces acting on it. Here's a basic derivation of each law:

1. Newton's First Law (Law of Inertia): This law states that an object at rest tends to stay at rest, and an object in motion tends to stay in motion, with the same speed and in the same direction, unless acted upon by a net external force. This is essentially the definition of inertia. The derivation of this law is more conceptual than mathematical. It's based on observation and the fundamental understanding of motion.

2. Newton's Second Law (F=ma): This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The derivation of this law is as follows:

   Let's assume 'F' is the net force applied on an object of mass 'm' and it produces an acceleration 'a' in the object. According to the law, F ∝ a (Force is directly proportional to acceleration) and F ∝ 1/m (Force is inversely proportional to mass). Combining these two, we get F ∝ a/m. To remove the proportionality, we introduce a constant of proportionality. Hence, F = k * a/m. The constant of proportionality 'k' is taken as 1 for simplicity and by convention. Hence, the equation becomes F = m*a.

3. Newton's Third Law (Action-Reaction Law): This law states that for every action, there is an equal and opposite reaction. This law is also more conceptual and is based on the fundamental understanding of forces. If object A exerts a force on object B, then object B must exert a force of equal magnitude in the opposite direction on object A.