# Mathematical Formulation Of Second Law Of Motion

We often observe, that if the same magnitude of force is used to push two blocks of wood, where one of the blocks is heavier than the other, the rate of change of position of the lighter block will be more than the heavier one. Similarly, when two same forces are applied to push a car and a bus, the car will have more acceleration compared to the bus. From these examples, it is clear that the acceleration gained by an object when subjected to the same magnitude of the force is a factor of the mass of the object. When a car is given a momentary jerk, it may not move from its initial position whereas when an extended and continuous force of the same magnitude is applied to the car, it experiences a displacement. With these examples, we can conclude that the impact produced by an object depends on its mass and velocity i.e., its momentum and the time rate at which the change in momentum is occurring. The second law of motion is used to validate this phenomenon. In this section, we shall learn about the formulation of the second law of motion.

## Newton’s Second Law of Motion

Newton’s second law of motion can be formally stated as follows: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.

## Second Law of Motion Formula

Let us consider an object of mass m, moving along a straight line with an initial velocity of u. Let us say, after a certain time t, with a constant acceleration, the final velocity becomes v. Here we see that, the initial momentum is:

$$\begin{array}{l}(p_1)=m \times u\end{array}$$

The final momentum,

$$\begin{array}{l}(p_2)=m \times v\end{array}$$

The change in momentum can be written as,

$$\begin{array}{l}(p_2) – (p_1) =(m \times v)- (m \times u)=m\times(v-u)\end{array}$$

As we know, the rate of change of momentum with respect to time is proportional to the applied force. The applied force,

$$\begin{array}{l}F \propto \frac{(m\times(v-u))}{t}\end{array}$$

Or,

$$\begin{array}{l}F\propto m \times a \end{array}$$

as acceleration (a) = rate of change of velocity with respect to time.

$$\begin{array}{l}F= k\times m \times a\end{array}$$

Above is the second law of motion formula.

### Notations Used In The Formula

• F is the force
• k is the constant of proportionality
• a is the acceleration

The SI units of mass and acceleration are kg and m.s-2 respectively.
So,

$$\begin{array}{l} Unit\,of\,Force = k \times ( 1\,kg) \times (1\, ms^{-2})\end{array}$$

The second law of motion gives us a method to measure the force acting on an object as a product of the mass of the object and the acceleration of the object which is the change in velocity with respect to time.

## Frequently Asked Questions – FAQs

Q1

### What are Newton’s laws of motion?

Newton’s laws of motion are three laws of classical physics that describe the connection between the motion of bodies and the forces acting on them.
Q2

### What are the three laws of motion in classical mechanics?

The law of inertia (Newton’s first law), the law of force and acceleration (Newton’s second law), and the law of action and reaction (Newton’s third law).
Q3

### Define Newton’s first law of motion.

Newton’s first law of motion states that an object at rest or uniform motion will continue to be at rest or uniform motion until and unless an external force acts on it.
Q4

### Define Newton’s second law of motion.

Newton’s second law of motion states that the acceleration of a body as generated by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the body’s mass.
Q5

### Define Newton’s third law of motion.

Newton’s third law of motion states that there is an equal and opposite reaction for every action.