Newton's Second Law of Motion: Definition, Formula, Derivation, and Applications

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 17^th 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:

Example 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.

Derivation of Newton's Second Law of Motion

Let'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 'X_0' and time 't_0​'.
• The car has a mass m₀​ and travels with a velocity 'v_0' .
• After being subjected to a force 'F' the car moves to point 1, defined by location 'X_1'​ and time 't_1' .
• The car’s mass and velocity change to 'm₁ ' and 'v_1​ ' after the force acts on it.
• Using Newton’s second law, we can determine the new values of 'm_1' and 'v_1'​ given the force 'F' .
• The equation for the force acting on the car is derived as:

F= m_1​v_1​ −m₀​v_0​/ t_1​−t_0​

• Assuming constant mass (which is a reasonable approximation for a car), the mass m_0​ 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(v_1​−v_0​)​/ t_1​−t_0​

• Acceleration is defined as the change in velocity divided by the change in time:

a= v_1​−v₀/ t_1​−t_0​

• 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

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.

Acceleration 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 (F_action) is equal and opposite to the reaction of object B on A (F_reaction).

Mathematically, this can be expressed as:

F_action = −F_reaction

• 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.

Related Reads,

• Newton's First Law of Motion
• Newton's Third law of Motion
• Equations of Motion
• Conservation of Momentum
• Planck's Constant
• Kinematic Equation of Motion
• Law of Inertia and Examples
• Mass and Inertia
• What is Acceleration?
• Motion in Physics