Force is equal to the rate of change of momentum. For a constant mass, force equals mass times acceleration.
Newton’s second law of motion, unlike the first law of motion pertains to the behaviour of objects for which all existing forces are unbalanced. The second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force.
Defining Newton’s Second Law of Motion
Newton’s second law states that the acceleration of an object depends upon two variables – the net force acting on the object and the mass of the object. The acceleration of the body is directly proportional to the net force acting on the body and inversely proportional to the mass of the body. This means that as the force acting upon an object is increased, the acceleration of the object is increased. Likewise, as the mass of an object is increased, the acceleration of the object is decreased.
Newton’s second law can be formally stated as,
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.
This statement is expressed in equation form as,
\(a=\frac{F_{net}}{m}\)
The above equation can be rearranged to a familiar form as
\(F=ma\)
Since force is a vector, Newton’s second law can be written as
\(\vec{F}=m\vec{a}\)
The equation shows that the direction of the total acceleration vector points in the same direction as the net force vector.
Deriving Newton’s Second Law
For Changing Mass
Let us assume that we have a car at a point (0) defined by location X_{0} and time t_{0}. The car has a mass m_{0} and travels with a velocity v_{0}. After being subjected to a force F, the car moves to point 1 which is defined by location X_{1} and time t_{1}. The mass and velocity of the car change during the travel to values m_{1} and v_{1}. Newton’s second law helps us determine the new values of m_{1} and v_{1} if we know the value of the acting force.
Taking the difference between point 1 and point 0, we get an equation for the force acting on the car as follows:
\(F=\frac{m_1v_1m_0v_0}{t_1t_0}\)Let us assume the mass to be constant. This assumption is good for a car because the only change in mass would be the fuel burned between point “1” and point “0”. The weight of the fuel is probably small relative to the weight of the rest of the car, especially if we only look at small changes in time. Meanwhile, if we were discussing the flight of a bottle rocket, then the mass does not remain constant and we can only look at changes in momentum.
For Constant Mass
For a constant mass, Newton’s second law can be equated as follows:
\(F=m\frac{v_1v_0}{t_1t_0}\)We know that acceleration is defined as the change in velocity divided by the change in time.
The second law then reduces to a more familiar form as follows: \(F=ma\)
The above equation tells us that an object will accelerate if it is subjected to an external force and the amount of force is directly proportional to the acceleration and inversely proportional to the mass of the object.

A net force ΣF is the sum of all forces acting on a body. More precisely, it is the vector sum of all forces acting on a body.
\(\sum F=30\,N20\,N=10\,N\)
\(\sum F=10\,N\,to \,the\, right\) 
Application of Second Law
The application of the second law of motion can be seen in identifying the amount of force needed to make an object move or to make it stop. Following are a few examples that we have listed to help you understand this point:
Kicking a ball
When we kick a ball we exert force in a specific direction, which is the direction in which it will travel. In addition, the stronger the ball is kicked, the stronger the force we put on it and the further away it will travel.
Pushing a cart
It is easier to push an empty cart in a supermarket than it is to push a loaded one. More mass requires more force to accelerate.
Two people walking
Among the two people walking, if one is heavier than the other then the one weighing heavier will walk slower because the acceleration of the person weighing lighter is greater.
Get a glimpse of Newton’s second law of motion being taught in BYJU’S classes.
Newton’s Second Law Solved Examples
Example 1:
If there is a block of mass 2kg, and a force of 5N is acting on it in the positive xdirection, and a force of 3N in the negative xdirection, then what would be its acceleration?
To calculate its acceleration, we first have to calculate the net force acting on it.
\(F_{net}\) = 5N – 3N = 2N
Mass = 2kg
∴ Acceleration = \( \frac 22 \) = 1 m/\(s^2\)
Example 2:
How much horizontal net force is required to accelerate a 1000 kg car at 4 m/s^{2}?
Solution:
Newton’s 2nd Law relates an object’s mass, the net force on it, and its acceleration:
Therefore, we can find the force as follows:
F_{net} = ma
Substituting the values, we get
1000 kg × 4 m/s^{2} = 4000 N
Therefore, the horizontal net force is required to accelerate a 1000 kg car at 4 m/s_{2} is 4000 N.
Newton’s second law is applied in daily life to a great extent. For instance, in Formula One racing, the engineers try to keep the mass of cars as low as possible. Low mass will imply more acceleration, and the more the acceleration, the chances to win the race are higher.
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Frequently Asked Questions – FAQs
How does Newton’s second law of motion apply to rockets?
We know that according to Newton’s second law of motion, force is a product of mass and acceleration. When a force is applied on the rocket, the force is termed as thrust. Greater the thrust, greater will be the acceleration. Acceleration is also dependent on the mass of the rocket. Lighter the rocket faster is the acceleration.
How does Newton’s second law apply to a car crash?
According to the definition of Newton’s second law of motion, force is the dot product of mass and acceleration. The force in a car crash is dependent either on the or the acceleration of the car. As the acceleration or mass of the car increases, the force with which car crash takes place will also increase.
What are Newton’s second law examples in everyday life?
Following are Newton’s second law examples in everyday life:
 Pushing a car is easier than pushing a truck with the same amount of force as the mass of the car is lesser than the mass of the truck.
 In golf game, acceleration of the golf ball is directly proportional to the force with which it is hit by the golf stick. Also, the force applied is inversely proportional to the mass of the ball.
What is the other name for Newton’s second law?
The other name for Newton’s second law is a law of force and acceleration.
What are some daily life examples of Newton’s second law of motion?
Newton’s second law of motion explains how force can change the acceleration of the object and how acceleration and mass of the same object are related. Therefore, in daily life, if there is any change in the acceleration of the object due to the applied force, then they are the examples of Newton’s second law.
 Acceleration of the rocket is due to the force applied known as thrust and is an example of Newton’s second law of motion.
 Another example of Newton’s second law is when an object falls down from a certain height, the acceleration increases because of the gravitational force.
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Superb explanation,understood the second law very well,prefer everyone to take byjus
perfect explanation