Newton's Second Law of Motion: Definition, Formula, Derivation, and Applications
Last Updated :
23 Jul, 2025
Newton's Second Law of Motion is a fundamental principle that explains how the velocity of an object changes when it is subjected to an external force. This law is important in understanding the relationship between an object's mass, the force applied to it, and its acceleration.
Here, we will learn about Newton's Second Law of motion, including its definition, example, formula, and derivation, and explore its real-life applications and more.
What is Newton's Second Law of Motion?
Newton's Second Law of Motion states that:
"The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass."
To explain Newton's Second Law of Motion further, consider that this law quantifies the impact of force on the motion of an object. If you apply a force to an object, it will accelerate in the direction of the force.
The amount of acceleration depends on the object's mass; heavier objects require more force to accelerate the same amount as lighter objects.
- Sir Isaac Newton was the one who proposed the Law of Motion in the 17th century.
- According to Newton's 2nd law of motion, the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass.
- Second Law of motion by Newton tells us about the motion of objects experiencing unbalanced forces.
- The second law of motion provides a relationship between the force and acceleration of any object in the universe.
Mathematical Expression of Newton's Second Law,
The second Law of Motion is also called the Quantitative Law of Motion because it quantitatively describes force.
In his mathematical formulation, Newton defined his Second Law of Motion as:
Force ∝ Change in Momentum / Change in the Time
F ∝ dp / dt
F ∝ m dv / dt = ma
where,
- F is Force
- dp is Change in Momentum
- dt is Change in Tme Taken
- p = mv
- a = dv/dt
Newton's Second Law- Examples
Newton's Second Law of Motion examples can be seen in various everyday scenarios:
Example 1: Pushing a Shopping Cart
Imagine you're pushing a shopping cart. The harder you push (more force), the faster it rolls (accelerates), right? That's the basic idea behind Newton's Second Law of Motion!
- Push or Pull (Force): This is anything that makes an object move or change its speed. It can be a kick, a throw, a car engine, or even gravity.
- Unbalanced: This means there's no other force pushing or pulling in the opposite direction. If someone else were pushing the cart back at the same time, it wouldn't move as fast, would it?
- Acceleration: This is how fast the object's speed is changing. It's not just about going faster, it can also mean slowing down or changing direction.
So, the Second Law says that the bigger the unbalanced force acting on an object, the greater its acceleration. And the heavier the object (think a full cart vs. an empty one), the less it will accelerate with the same force.
Example 2: Car Acceleration
A car's acceleration depends on the engine's force and the car's mass. A sports car (less mass) accelerates faster than a truck (more mass) when the same amount of force is applied.
Example 3: Catching a Ball
- On the cricket ground, the fielder pulls his hands in the backward direction to catch the fast-approaching cricket ball. This reduces the momentum of the ball and induces a delay. When the ball comes into the hand of the fielder and comes to a halt, the momentum of the ball is reduced to zero.
- In case, the ball stops suddenly the momentum reaches '0' in an instant time frame.
- There is a quick rate of change in momentum due to which the player's hand may get injured. Therefore, pulling the hand backward a fielder induces a delay to the change of momentum to become zero, to prevent injury.
The image given below shows a fielder catching a ball and taking their hand backward to prevent injury:
Fielder pulling hand backwardsExample 4: Sudden Change in a Vehicle's Velocity
In the event of a sudden change in a vehicle's velocity, as a result of braking or an accident, the passengers tends to be pushed in the forward direction and may get fatal injuries. In such a case, the change of momentum is reduced to zero by seat belts. Through their stretch, seat belts prolong the time it takes for passengers' momentum to reach zero, reducing the risk of injury in a crash.
Sudden Change in a Vehicle's VelocityLet's derive the 2nd law of motion.
The rate of change in the momentum of a body is directly proportional to the applied force and occurs in the force's direction. Newton's first law explains the concept of inertia, while Newton's Second Law provides a numerical relationship between force, mass, and acceleration."
Consider a body with instantaneous velocity \vec{v} and momentum \vec{p} given by:
\vec{p} = m\vec{v}
Since, according to the second law of motion,
\vec{F}∝\dfrac{d\vec{p}}{dt}
Where \vec{F} is the force acting on the object.
Also, since the momentum is defined as,
\vec{p} = m\vec{v}
Therefore, the previous equation becomes,
\vec{F}\alpha \dfrac{d(m\vec{v})}{dt}
\vec{F}=k\dfrac{d(m\vec{v})}{dt}
Where k is the constant of proportionality.
As the mass m of a body can be considered to be a constant quantity so derivative is applicable to the velocity of the body as shown below,
\vec{F}=km\dfrac{d(\vec{v})}{dt}
It is known that the time rate change of velocity of the body is termed as its acceleration i.e.
\vec{a}=\dfrac{d\vec{v}}{dt}
Therefore,
\vec{F}=km\vec{a}
The units of force are also chosen such that ‘k' equals one.
As a result, if a unit force is selected to be the force causing a unit acceleration in a unit mass, i.e.,
F = 1 N, m = 1 kg and a = 1 ms-2. This implies, k = 1.
Thus, Newton’s second law of motion in mathematical form is given as
\bold{\vec{F}\ =m\vec{a}}
That is, the applied force of a body is defined as the product of its mass and acceleration. Hence, this provides us with a measure of the force.
If F = 0, we get a = 0.
This is similar to Newton's first law of motion. That is, if there is no net external force, there will be no change in state of motion, implying that its acceleration is zero.
Deriving Newton’s Second Law
For Changing Mass:
- Assume a car starts at point '0' with location 'X0' and time 't0'.
- The car has a mass m0 and travels with a velocity 'v0' .
- After being subjected to a force 'F' the car moves to point 1, defined by location 'X1' and time 't1' .
- The car’s mass and velocity change to 'm1 ' and 'v1 ' after the force acts on it.
- Using Newton’s second law, we can determine the new values of 'm1' and 'v1' given the force 'F' .
- The equation for the force acting on the car is derived as:
F= m1v1 −m0v0/ t1−t0
- Assuming constant mass (which is a reasonable approximation for a car), the mass m0 does not change significantly during travel.
- The only change in mass would be the fuel burned, which is small compared to the car's total mass, especially for short timescales.
- However, for a system like a bottle rocket where the mass changes significantly, we would need to focus on changes in momentum, not just velocity.
For Constant Mass:
- For a constant mass, Newton's second law can be written as:
F= m(v1−v0)/ t1−t0
- Acceleration is defined as the change in velocity divided by the change in time:
a= v1−v0/ t1−t0
- Substituting the definition of acceleration into the equation, we get the familiar form of Newton's second law:
F=ma
- This equation shows that an object will experience acceleration when subjected to an external force.
- The force applied is directly proportional to the acceleration of the object and inversely proportional to its mass.
Also Read, What is net force?
Applications of Newton's Second Law of Motion
Applications of Newton's Second Law of Motion are:
Pushing an Object
It's no secret: pushing a light thing is way easier than pushing a heavy one, even if they look similar! This observation comprehends Newton's Second Law of Motion.
Kicking a football changes its direction and also changes its velocity. It can increase or decrease the velocity of football. The force applied by the footballer is responsible for the change that the ball produces. Thus, Newton's Second Law of Motion also holds true in this case.
A person kicking a footballAcceleration of Rocket
The acceleration of the rocket is due to the force applied called Thrust. This force makes the rocket go up with an acceleration of 'a' where 'a' is Thrust divided by mass.
Newton's 2nd Law as Real Law of Motion
- Newton's second law of motion is a cornerstone of classical mechanics, describing the relationship between the motion of an object and the forces acting upon it.
- Mathematically expressed as F = ma, this law encapsulates not only the fundamental principle of force and acceleration but also embodies the concepts inherent in Newton's first and third laws of motion.
- In this section, we will delve into the mathematical proof that shows how Newton's second law incorporates the essence of both the law of inertia (Newton's first law) and the law of action and reaction (Newton's third law).
First Law in Second Law:
If there is no net external force (F=0), then ma=0, which implies a=0, because 'm' can never be zero. This aligns with the first law, indicating that an object will remain at rest or move with a constant velocity when no force is applied.
Third Law in Second Law:
- Consider two objects, A and B, interacting with each other. According to the third law, the action of object A on B (Faction) is equal and opposite to the reaction of object B on A (Freaction).
Mathematically, this can be expressed as:
Faction = −Freaction
- In the second law, the force (F) acting on an object is the result of the interaction between that object and another object.
- The acceleration (a) that results from this force is determined by the mass (m) of the object. So, the third law is embedded in the second law through the interactions of objects and the resulting forces.
Common Misconceptions about Newton's Second Law
Some common misconceptions about Newton's second law of motion include:
- It only applies to objects in motion.
- It does not consider the direction of the force.
- It can be used interchangeably with Newton's first and third laws.
Solved Examples -Newton's Second Law of Motion
Example 1: If a bullet of mass 40 gm is shot from an Assault Riffle that has an initial velocity of 80 m/s the mass of the Assault Riffle is 15 kg. What is the initial recoil velocity of the Assault Riffle?
Given,
Mass of bullet (m1) = 40 gm or 0.04 kg
Mass of the Assault Riffle (m2) = 15 kg
Initial velocity (v1) = 80 m/s.
Therefore, according to the law of conservation of momentum,
0 = 0.04 × 80 + 15 × v
⇒ 15 × v = -3.2
⇒ v = -3.2 / 15
⇒ v = -0.21 m/s
Example 2: If an object of mass 20 kg is moving with a constant velocity of 8 m/s on the frictionless ground. What will be the force required to keep the body moving with the same velocity?
Given,
Mass of the object (m) = 20 kg.
Acceleration of the object (a) = 0 (as object is moving constantly).
Applied force is given as,
F = m × a
⇒ F = 20 kg × 0
⇒ F = 0 N
Example 3: If a heavy truck weighing 2000 kg is running with some velocity. If the driver applies brakes and is brought to rest, after application of brakes the heavy truck goes about 20 m when the average resistance being offered to it is 4000 N. What will be the velocity of the heavy truck engine?
Given:
Mass of truck (m) = 2000 kg
Resistance (F) = - 4000 N [negative as stopping force is applied]
Distance traveled after applying brakes (s) = 20 m.
Final velocity (v) = 0 m/s [as the heavy truck was brought to rest]
To find the initial velocity (?) of the truck, we'll use the equations of motion. The equation relating initial velocity (?) , final velocity (v) , acclearation (a), distance (s) :
v2 = u2 + 2as
Since the truck comes to rest, its final velocity (v) is 0 , We want to find the initial velocity (?).
0 = u2 + 2(-2) x 20
0 = u2 -80
u2 = 80
u = √80
u = 8.94m/s
Example 4: A mini truck of 2500 kg with a velocity of v runs head-on with a big truck of 5000 kg with a velocity of −v. Which truck will experience the greater force? Which experiences the greater acceleration?
According to the Newton's second law of motion,
F = ma
⇒ a = F / m
Mini truck and the big truck experience equal and opposite forces. But since the mini truck has a smaller mass it will experience greater acceleration than the big truck.
Hence, the truck with greater mass's acceleration will be decreased.
Example 5: What will be the net force needed to accelerate a 1000 kg car at 8 m/s2?
Given,
Acceleration of car (a) = 8 m/s2
Mass of car (m)= 1000 kg
Therefore, using the formula for the applied force as,
F = m × a
⇒ F = 1000 kg × 8 m/s2
⇒ F = 8000 N
Example 6. If a net force of 12 N is applied to a 1 kg object, what will be the acceleration of the object?
Given,
Force applied (F) = 12 N.
Mass (m) = 1 kg.
Therefore, using the formula for the applied force as,
F = m × a
⇒ a = F / m
⇒ a = 12 N / 1 kg
⇒ a = 12 m/s2
Practice Problems - Newton's Second Law of Motion
Problem 1: A 5 kg object experiences a force of 20 N. Calculate the acceleration of the object.
Problem 2: A car with a mass of 1,200 kg accelerates at a rate of 3 m/s². What is the force applied to the car?
Problem 3: If you push a 50 kg box with a force of 200 N, what will be the acceleration of the box?
Problem 4: An astronaut with a mass of 70 kg is on the Moon, where gravity is about 1/6th that of Earth's. Calculate the astronaut's weight on the Moon and the force required to accelerate them at 5 m/s².
Problem 5: A rocket with a mass of 1,000 kg is launched into space. If it experiences a constant thrust force of 10,000 N, what will be its acceleration?
Conclusion
Newton's second law of motion states that the force acting on an object is equal to its mass multiplied by its acceleration, or F=ma. This implies that the acceleration of an object increases with the applied force and decreases with the object's mass.
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First Law of ThermodynamicsFirst Law of Thermodynamics adaptation of the Law of Conservation of Energy differentiates between three types of energy transfer: Heat, Thermodynamic Work, and Energy associated with matter transfer. It also relates each type of energy transfer to a property of a body's Internal Energy. The First L
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Second Law of ThermodynamicsSecond Law of Thermodynamics defines that heat cannot move from a reservoir of lower temperature to a reservoir of higher temperature in a cyclic process. The second law of thermodynamics deals with transferring heat naturally from a hotter body to a colder body. Second Law of Thermodynamics is one
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Thermodynamic CyclesThermodynamic cycles are used to explain how heat engines, which convert heat into work, operate. A thermodynamic cycle is used to accomplish this. The application determines the kind of cycle that is employed in the engine. The thermodynamic cycle consists of a series of interrelated thermodynamic
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Thermodynamic State Variables and Equation of StateThe branch of thermodynamics deals with the process of heat exchange by the gas or the temperature of the system of the gas. This branch also deals with the flow of heat from one part of the system to another part of the system. For systems that are present in the real world, there are some paramete
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Enthalpy: Definition, Formula and ReactionsEnthalpy is the measurement of heat or energy in the thermodynamic system. It is the most fundamental concept in the branch of thermodynamics. It is denoted by the symbol H. In other words, we can say, Enthalpy is the total heat of the system. Let's know more about Enthalpy in detail below.Enthalpy
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State FunctionsState Functions are the functions that are independent of the path of the function i.e. they are concerned about the final state and not how the state is achieved. State Functions are most used in thermodynamics. In this article, we will learn the definition of state function, what are the state fun
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Carnot EngineA Carnot motor is a hypothetical motor that works on the Carnot cycle. Nicolas Leonard Sadi Carnot fostered the fundamental model for this motor in 1824. In this unmistakable article, you will find out about the Carnot cycle and Carnot Theorem exhaustively. The Carnot motor is a hypothetical thermod
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Heat Engine - Definition, Working, PV Diagram, Efficiency, TypesHeat engines are devices that turn heat energy into motion or mechanical work. Heat engines are based on the principles of thermodynamics, specifically the conversion of heat into work according to the first and second laws of thermodynamics. They are found everywhere, from our cars, power plants to
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Wave and Oscillation
Introduction to Waves - Definition, Types, PropertiesA wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities in physics, mathematics, and related subjects, commonly described by a wave equation. At least two field quantities in the wave medium are involved in physical waves. Periodic waves occur when variables o
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Wave MotionWave Motion refers to the transfer of energy and momentum from one point to another in a medium without actually transporting matter between the two points. Wave motion is a kind of disturbance from place to place. Wave can travel in solid medium, liquid medium, gas medium, and in a vacuum. Sound wa
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OscillationOscillations are defined as the process of repeating vibrations of any quantity about its equilibrium position. The word “oscillation†originates from the Latin verb, which means to swing. An object oscillates whenever a force pushes or pulls it back toward its central point after displacement. This
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Oscillatory Motion FormulaOscillatory Motion is a form of motion in which an item travels over a spot repeatedly. The optimum situation can be attained in a total vacuum since there will be no air to halt the item in oscillatory motion friction. Let's look at a pendulum as shown below. The vibrating of strings and the moveme
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Amplitude FormulaThe largest deviation of a variable from its mean value is referred to as amplitude. It is the largest displacement from a particle's mean location in to and fro motion around a mean position. Periodic pressure variations, periodic current or voltage variations, periodic variations in electric or ma
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What is Frequency?Frequency is the rate at which the repetitive event that occurs over a specific period. Frequency shows the oscillations of waves, operation of electrical circuits and the recognition of sound. The frequency is the basic concept for different fields from physics and engineering to music and many mor
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Amplitude, Time Period and Frequency of a VibrationSound is a form of energy generated by vibrating bodies. Its spread necessitates the use of a medium. As a result, sound cannot travel in a vacuum because there is no material to transfer sound waves. Sound vibration is the back and forth motion of an entity that causes the sound to be made. That is
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Energy of a Wave FormulaWave energy, often referred to as the energy carried by waves, encompasses both the kinetic energy of their motion and the potential energy stored within their amplitude or frequency. This energy is not only essential for natural processes like ocean currents and seismic waves but also holds signifi
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Simple Harmonic MotionSimple Harmonic Motion is a fundament concept in the study of motion, especially oscillatory motion; which helps us understand many physical phenomena around like how strings produce pleasing sounds in a musical instrument such as the sitar, guitar, violin, etc., and also, how vibrations in the memb
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Displacement in Simple Harmonic MotionThe Oscillatory Motion has a big part to play in the world of Physics. Oscillatory motions are said to be harmonic if the displacement of the oscillatory body can be expressed as a function of sine or cosine of an angle depending upon time. In Harmonic Oscillations, the limits of oscillations on eit
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Sound
Production and Propagation of SoundHave you ever wonder how are we able to hear different sounds produced around us. How are these sounds produced? Or how a single instrument can produce a wide variety of sounds? Also, why do astronauts communicate in sign languages in outer space? A sound is a form of energy that helps in hearing to
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What are the Characteristics of Sound Waves?Sound is nothing but the vibrations (a form of energy) that propagates in the form of waves through a certain medium. Different types of medium affect the properties of the wave differently. Does this mean that Sound will not travel if the medium does not exist? Correct. It will not, It is impossibl
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Speed of SoundSpeed of Sound as the name suggests is the speed of the sound in any medium. We know that sound is a form of energy that is caused due to the vibration of the particles and sound travels in the form of waves. A wave is a vibratory disturbance that transfers energy from one point to another point wit
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Reflection of SoundReflection of Sound is the phenomenon of striking of sound with a barrier and bouncing back in the same medium. It is the most common phenomenon observed by us in our daily life. Let's take an example, suppose we are sitting in an empty hall and talking to a person we hear an echo sound which is cre
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Refraction of SoundA sound is a vibration that travels as a mechanical wave across a medium. It can spread via a solid, a liquid, or a gas as the medium. In solids, sound travels the quickest, comparatively more slowly in liquids, and the slowest in gases. A sound wave is a pattern of disturbance caused by energy trav
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How do we hear?Sound is produced from a vibrating object or the organ in the form of vibrations which is called propagation of sound and these vibrations have to be recognized by the brain to interpret the meaning which is possible only in the presence of a multi-functioning organ that is the ear which plays a hug
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Audible and Inaudible SoundsWe hear sound whenever we talk, listen to some music, or play any musical instrument, etc. But did you ever wondered what is that sound and how is it produced? Or why do we hear to our own voice when we shout in a big empty room loudly? What are the ranges of sound that we can hear? In this article,
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Explain the Working and Application of SONARSound energy is the type of energy that allows our ears to sense something. When a body vibrates or moves in a ‘to-and-fro' motion, a sound is made. Sound needs a medium to flow through in order to propagate. This medium could be in the form of a gas, a liquid, or a solid. Sound propagates through a
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Noise PollutionNoise pollution is the pollution caused by sound which results in various problems for Humans. A sound is a form of energy that enables us to hear. We hear the sound from the frequency range of 20 to 20000 Hertz (20kHz). Humans have a fixed range for which comfortably hear a sound if we are exposed
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Doppler Effect - Definition, Formula, ExamplesDoppler Effect is an important phenomenon when it comes to waves. This phenomenon has applications in a lot of fields of science. From nature's physical process to planetary motion, this effect comes into play wherever there are waves and the objects are traveling with respect to the wave. In the re
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Doppler Shift FormulaWhen it comes to sound propagation, the Doppler Shift is the shift in pitch of a source as it travels. The frequency seems to grow as the source approaches the listener and decreases as the origin fades away from the ear. When the source is going toward the listener, its velocity is positive; when i
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Electrostatics
ElectrostaticsElectrostatics is the study of electric charges that are fixed. It includes an study of the forces that exist between charges as defined by Coulomb's Law. The following concepts are involved in electrostatics: Electric charge, electric field, and electrostatic force.Electrostatic forces are non cont
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Electric ChargeElectric Charge is the basic property of a matter that causes the matter to experience a force when placed in a electromagnetic field. It is the amount of electric energy that is used for various purposes. Electric charges are categorized into two types, that are, Positive ChargeNegative ChargePosit
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Coulomb's LawCoulomb’s Law is defined as a mathematical concept that defines the electric force between charged objects. Columb's Law states that the force between any two charged particles is directly proportional to the product of the charge but is inversely proportional to the square of the distance between t
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Electric DipoleAn electric dipole is defined as a pair of equal and opposite electric charges that are separated, by a small distance. An example of an electric dipole includes two atoms separated by small distances. The magnitude of the electric dipole is obtained by taking the product of either of the charge and
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Dipole MomentTwo small charges (equal and opposite in nature) when placed at small distances behave as a system and are called as Electric Dipole. Now, electric dipole movement is defined as the product of either charge with the distance between them. Electric dipole movement is helpful in determining the symmet
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Electrostatic PotentialElectrostatic potential refers to the amount of electrical potential energy present at a specific point in space due to the presence of electric charges. It represents how much work would be done to move a unit of positive charge from infinity to that point without causing any acceleration. The unit
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Electric Potential EnergyElectrical potential energy is the cumulative effect of the position and configuration of a charged object and its neighboring charges. The electric potential energy of a charged object governs its motion in the local electric field.Sometimes electrical potential energy is confused with electric pot
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Potential due to an Electric DipoleThe potential due to an electric dipole at a point in space is the electric potential energy per unit charge that a test charge would experience at that point due to the dipole. An electric potential is the amount of work needed to move a unit of positive charge from a reference point to a specific
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Equipotential SurfacesWhen an external force acts to do work, moving a body from a point to another against a force like spring force or gravitational force, that work gets collected or stores as the potential energy of the body. When the external force is excluded, the body moves, gaining the kinetic energy and losing a
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Capacitor and CapacitanceCapacitor and Capacitance are related to each other as capacitance is nothing but the ability to store the charge of the capacitor. Capacitors are essential components in electronic circuits that store electrical energy in the form of an electric charge. They are widely used in various applications,
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