Newton’s second law: Formula, Derivation & Solved examples

Newton’s second law of motion is proposed to highlight what happens when the net external force acts to accelerate a body. Developed in 1666, the 3 laws of motion of Sir Issac Newton were discovered to observe the object's motion in an inertial reference frame. He found that in motions of an inertial frame where, during the absence of an external force, objects either stay in rest or stay in motion. The second law, in particular, is only restricted to the inertial frame. Newton’s laws of motion have a huge contribution to Modern physics.

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Newton’s Second Law

The statement of the second law of motion expressed by Newton is as follows.

“The rate of change of momentum of a body is directly proportional to the applied force & takes place in the direction in which the force acts.” 

Illustration of Newton’s Second Law

In simple terms, the second law states that the acceleration of an object depends upon the force that acts on the object and the mass of the said object. Here, the acceleration is directly proportional to the acting force while it is inversely proportional to the mass of the object. Now let us understand what momentum is.

Laws of Motion Video Lecture

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What is Momentum? 

Momentum of a body is formulated by the product of its mass and its velocity. It is a vector quantity since it has both the magnitude(|mv|) and direction due to velocity. Momentum can be represented as,

p = mv

Where,

m → Mass 

v → Velocity 

A common example of momentum is,

Momentum Example

Imagine a heavy-loaded and a light-weighted truck parked on the road.

While moving at the same speed, the heavy-loaded truck needs more external force than the light-weighted to bring them to the same speed. Similarly,the heavy-loaded truck needed greater stopping force due to its greater momentum.

Read More: Radius of Gyration

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Newton’s Second Law Formula and Derivation

The formula for Newton’s second law of motion is given by,

F = ma

Where,

F → External Force 

m → Mass 

a → Acceleration

Analysing the statement of the second law, 

If an external force F is acting on a body of mass m. Say, its velocity changes from v to (v+Δv) in a time interval Δt.

The momentum also changes from mv to mΔv.

∴ F∝ pt 

or, 

F=c.\(\frac{\bigtriangleup p}{\bigtriangleup t}\) [c is a constant]

If we take the limit Δt → 0, \(\frac{\bigtriangleup p}{\bigtriangleup t}\) becomes \(\frac{dp}{dt}\)

Now,  

F = c.\(\frac{dp}{dt}\)= c.\(\frac{d(mv)}{dt}\)= c.m.\(\frac{d(v)}{dt}\)= c.m.a 

Where,

\(\frac{d(v)}{dt}\) → a → acceleration (rate of change in velocity)

Thus, we derived,

F∝ ma 

or 

F = ma [c=1]

Note: It is possible to derive from the rate of change in momentum of a body and vice-versa.

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Units of Measurement

The SI unit of force is newton(N). As force is the product of mass and acceleration, we can derive its unit as follows.

Unit of mass kg (kilogram) 

Unit of accelerationmeter/second² (m/s²)

1N = 1kg.m/s²

The CGS unit of force is dyne. It is expressed as,

1dyn(dyne) = 1g.cm/s²

The Dimensional formula of Newton’s Second law is given by, 

F = ma = MLT^-2

It is important to note that force moves along the direction of acceleration of the body.

Read More: Difference between Force and Pressure

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Conditions of Second Law

• For being a vector, there are 3 equivalent equation,

F_x=ma_x ; F_y=ma_y ;F_z=ma_z

• When force isn’t parallel with the velocity of the body and makes an angle θ with each other,

1. The horizontal component of velocity changes.
2. Normal or perpendicular(?) remains the same.

• When F=0 then a=0. It proves the first law of motion.
• The law is applicable for a single point particle. But for a rigid body or system of particles in general, we have to calculate the acceleration of the center of mass.
• The second law is an instant relation between F and a.

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Real-life applications of Second Law of Motion

• While riding a bicycle, when we peddle, we provide an external force to it. This force creates momentum in the bicycle and it moves in the same direction.

Example of Newton’s second law

• In race cars, the mass of a car is kept low. Due to the low mass, the momentum becomes less and the acceleration increases. Thus, it helps to increase the chance of winning a race with high acceleration.
• In a football match, when a player kicks the ball, he exerts a force in a specific direction. The stronger he kicks, the further it will go.
• While catching a ball, cricketers hold their hands back. It gives the ball time to decrease its speed. Thus, by applying less force, they can catch the ball easily.

Cricketer catching the ball

• If two people walk together. If one of them weighs lighter, he will walk faster. Though both of them are walking with the same force, the momentum of the lighter person is less.  

Read More: Displacement Vector

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Things to Remember

• Newton’s second law of motion is only applicable for an inertial frame.
• The formula to ponder on for Newton’s second law of motion is F = ma [where F=external force, m=mass, a=acceleration]
• As of Newton's second law, external force on a body is equal to the rate of change of momentum of a body.
• Momentum is a vector quantity.
• We can prove the first law using the second law.
• For a system of particles or rigid bodies, take the acceleration of the centre of mass.

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