Newton’s Second Law: Definition, Units, Applications & Examples

Table of Contents

Introduction

Newton’s Second Law of Motion is one of the three fundamental laws Sir Issac Newton has formulated. This law shifted physics from qualitative (philosophical ideas) to quantitative science. It tells us the quantitative view of one quantity that actually changes the state of motion: the net force. Unlike Newton’s First Law, the Second Law explains how and why objects speed up, slow down, or turn. The Second Law of Motion provides the detailed quantitative relation between force, mass, and acceleration. In its simplest form, \( F=m a\) : more force means more acceleration; more mass means less acceleration for the same push. The measurable change a force produces depends on the net external force and its direction, which together set the object’s acceleration. Some everyday examples are — cars pulling away from a stop, balls kicked harder flying faster, and rockets steering in space. Use this law to predict motion, solve problems, and make sense of everyday physics.

Statement of the Law

⚖️ Statement of Newton’s Second Law
  • Newton’s Second Law of Motion: The acceleration of an object is directly proportional to the net external force and inversely proportional to its mass; the acceleration has the same direction as the net force.
    Constant-mass form:
  • In the case of constant mass: the net external force is given as mass times acceleration. \( \sum \vec F = m \vec a \)
    • General (momentum) form:
  • The net external force on a body equals the time rate of change of its momentum and points in the same direction. \( \sum \vec F = \dfrac{d\vec p}{dt}=m\dfrac{d\vec v}{dt}=m{\vec a} \)
Conditions:
  • All forms above assume inertial reference frames: (non-accelerating frames).

We have learned that when the forces are balanced, the total vector sum of the forces is zero, i.e., \(\sum {\vec F} =0\); and in the case of unbalanced forces it isn’t, i.e., \(\sum {\vec F} \neq 0\), which means we have acceleration in the object. This left us with the two different and important cases:

  • The acceleration depends directly upon the “net force”.
  • The acceleration depends inversely upon the object’s mass.

Mathematical Form and Units

Mathematically, Newton’s Second Law was derived from the definition of acceleration and the momentum of an object. The equation is given by:

\[\sum {\vec F}=m{\vec a}\tag{1}\]

where, \(\sum {\vec F}\) is the net external force (net sum of all pushes and pulls), \(m\) is the mass of an object, and \({\vec a}\) is the acceleration of an object. The equation \((1)\) clearly explains that the heavier object will have a less acceleration and when the mass is less the acceleration of an object will be higher, given the applied external force is constant on an object.

Notes: Use the vector form when direction matters (it usually does). If mass changes (like rockets), use the momentum form \(\sum {\vec F} = \dfrac{d\vec p}{dt}\). Always identify the system, draw the forces, add them as vectors, and then apply \(\sum {\vec F} = m {\vec a}\).

Units

In SI units:

  • Force, \((F)\), is measured in Newton (N or (kg.m/s\(^2\)).
  • Mass, \((m)\), is measured in kilograms (kg).
  • Acceleration, \((a)\), is measured in metre/second square (m/s\(^2\)).

In cgs units:

  • Force, \((F)\), is measured in dyne (dyne).
  • Mass, \((m)\), is measured in grams (gm).
  • Acceleration, \((a)\), is measured in centimetre/second square (cm/s\(^2)\).

Component form and Coordinates

We see the force equation is usually written in the scalar form, however, the force and acceleration both are vector quantities. They both have magnitude and direction. We have to always remember that, the direction of force is as much important as its magnitude.

For e.g. If a car is being pulled by a towing truck with a force of 400N to the east and at the same time, another towing truck is pulling it with force 500N to the west, the net force on the car is given by the difference between the forces with proper direction where it will move. 100 N towards the west.

The process involved in calculating the exact force acting on any system is done by following the processes involved, first break the forces into its vector components (e.g., horizontal and vertical forces) and analyze them using vector addition. While doing this process we make a free-body diagram of the system and all the forces acting on the system and lis the forces along the chosen axes and then sum them. Mathematically,

\[\sum {\vec F}_x=m{\vec a}_x, \sum {\vec F}_y=m{\vec a}_y, \sum {\vec F}_z=m{\vec a}_z \tag{2}\]

Workflow
  • Draw a free-body diagram and choose axes that match the motion/constraints.
  • Resolve the forces step wise into components along those axes.
  • Take care while writing the force equation for each axis and use clear sign convention.
  • Solve for unknowns (accelerations, tension, friction, etc.).

Free-body diagram

A free-body diagram (FBD) is a clean sketch of any object (chosen system) with all external forces acting on the object, shown as labelled arrows —e.g., weight \(mg\), normal \(N\), tension \(T\), friction \(f\), drag, or thrust. The initial step for a free body diagram is to pick axes (often along/normal to a surface), set a sign convention, and, if needed, resolve forces into components. We should pay attention in picking up only the external forces acting on the object and avoid strongly to double count the force and its components, and keep vectors from the contact point or center with correct directions. FBDs help to turn a word problem into math by feeding Newton’s Second Law component-wise: \(\sum F_x =ma_x, \sum F_y =ma_y\). The basic steps are:

  • Isolate the object
  • Draw and label all forces
  • Choose axes
  • Resolve components
  • Write \(\sum F = ma\) along each axis and solve.

Derivation of Newton’s Second Law of Motion

Newton’s initial idea for this law was basically for momentum. Momentum is generally a “quantity of motion” given as,

\[\sum {\vec p} = m{\vec v} \tag{3}\]

Newton’s key idea was that Forces don’t act on position directly — they change the momentum. In the Principia Mathematica (1687), the law is stated as;

\[\sum {\vec F} = \dfrac{d\vec p}{dt} \tag{4}\]

Constant–Mass Case:
i.e., the net external force equals the time rate of change of momentum (and points towards the same direction). When the mass of an object is constant. Using eqn: (3) and (4);

\[\sum {\vec F} = m\dfrac{d\vec v}{dt} = m\vec a : \left[\dfrac{d\vec v}{dt}=\vec a\right]\tag{5}\]
Variable–Mass Case (e.g., rockets):
\[\sum {\vec F} = \dfrac{d\vec p}{dt} = v\dfrac{dm}{dt} + m\dfrac{dv}{dt} \tag{6}\]

Additionally, if the force changes over time, it will also change the momentum. If a force acts for a small time interval, \(\Delta t\)

\[\sum {\vec F}\cdot \Delta t = \Delta \vec p \tag{7}\]

This explains why airbags and crumple zones increase stopping time to reduce force.

Applications and Significance

Vehicle Dynamics & Safety

In the automobile industry, Newton’s Second law is used to calculate the braking/acceleration forces, stopping distance, and tune the airbags/seatbelts. It is also used to understand the impact of the collision to understand the impact ability of the vehicles. The exact calculation of the force and acceleration helps engineers to find all the required information for the utmost safety and accuracy of the vehicles.

Rocket Launching

During a rocket launch, we all are aware that as the fuel keeps on burning it priovides the required force for launching of the rocket. The exhaust is generated by rapidly burning the fuel and in return, the rocket moves in upward directionin. It generally uses the momentum form of the Newton’s Second law of motion, \(\sum {\vec F}=\dfrac {d {\vec p}}{dt}\) for variable mass systems; expelling mass at exhaust speed \(v_e \) creates thrust, \(F_{thrust}=\dot m v_e\) (with \(\dot m >0\) as mass flow rate).

Sports Science

Atheletes uses Newton’s Second Law of Motion to generate greater acceleration by applying larger net forces or reducing mass in motion — as mass is inversely proportional to the acceleration. Coaches use impulse \(\left(\int F dt \right)\) to increase change in momentum: longer contact time or higher force makes a ball leave faster or a sprinter launch harder. Equipment design (rackets, shoes, bats) tunes mass and stiffness so the same swing force yields higher acceleration and better control.

Misconception About Newton’s Second Law

  • Mass and Weight are Same Quantity: Its quite easy to get confuse with mass and weight. Mass is defined as the total amount to matter contained in the body. Weight is defined as the total force exerted on that object by the gravitational force. The SI unit of mass in kilogram (kg) and that of weight is Newton (N).
  • Bigger force means higher speed: Force sets the acceleration, not speed. A brief big force gives a big change in speed; once the force stops, acceleration is zero and speed stays whatever it reached.
  • No force no motion: An object can keep moving at a steady speed in a straight line without any net force — changes in speed or direction requries the force. Take an example of space; a probe coasts for years, but on earth friction and air drag usually hide this by slowing things down. This is also well explained in Newton’s First Law, as the object tends to be in state unless and external force is applied on it.
  • Acceleration and Velocity are same: Velocity is defined as the rate of change of displacement with time — it tells how fast and in what direction an object moves. Velocity is a vector quantity but its magnitude is always positive. Acceleration whereas is the rate of change of velocity with time — how quickly speed and/or direction changes. Acceleration can be both negative and positive. Negative acceleration is also called as ‘retardation’.

Importance of Second Law and Non-inertial Frames

Generally, all of the Newton’s Laws of motion are formulated for inertial frames–observers moving at constant velocity. In this setting, the Second Law uses the net external force (the vector sum of all forces) to determine acceleration, calculate force.

  • First Law (Inertia): An object keeps its current state–rest or straight-line, constant-speed motion if the net force acting on it is zero
  • Second Law (Dynamics): When a non-zero force acts, it sets both the size and direction of the object’s acceleration: \(\sum {\vec F}=m{\vec a}\).
  • Third Law (Action-Reaction): Forces comes in pair, equal and oppsite pairs on different bodies; by contrast, the Second Law focuses on one body, summing only the external forces on it.
Outside inertial frames

In accelerating or rotating frames (non-inertial), you must introduce inertial/fictitious forces to use the Second Law consistently.

  • Centrifugal force: Appears to push outward in a rotating frames, though it has no couterpart in an intertial frame.
  • Coriolis force: Acts on moving objects within a rotating system (e.g., large-scale wind patterns on Earth), altering their paths relative to that frame.

Summary

Newton’s Second Law is a practical toolkit for solving real-world problems–from everyday motion to engineering design. We can model cars, bridges, rockets, and many more with confidence, provided we first identify the forces correctly and understand momentum. The approach scales well: it’s simple enough for classroom examples yet powerful for advanced analyses and simulations. Much of gravitational motion in spaceflight and astronomy is competed within this same framwork. Ongoing research builds on it to model complex, multi-body and high-precision systems across aerospace and beyond. In that sense, it remains one of the core pillars on which modern physics and engineering rest.

References

  1. Halliday, D., Resnick, R. and Walker, J. (2014) Fundamental of Physics. 10th Edition, Wiley and Sons, New York.
  2. Newton, Isaac, 1642-1727. Newton’s Principia : the Mathematical Principles of Natural Philosophy. New-York :Daniel Adee, 1846.
  3. https://byjus.com/physics/newtons-second-law-of-motion-and-momentum/
  4. https://www.britannica.com/science/Newtons-laws-of-motion/Newtons-second-law-F-ma
  5. https://www.physicsclassroom.com/class/newtlaws/lesson-3/newton-s-second-law
  6. https://sciencenotes.org/newtons-second-law-of-motion/
  7. https://www.sciencefacts.net/newtons-second-law.html