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

Last Updated : 31 Jan, 2024

Newton’s Second Law of Motion says that for an object under the influence of unbalanced forces, the acceleration of the object is directly proportional to the force applied.

In this article, we will learn about the Second law of motion by Newton, including its definition, example, formula, derivation, and applications. We will also explore some numerical problems and FAQs on the Second Law of motion.

## What is the Second Law of Motion?

Newton’s Second Law of Motion states that:

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

• 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.

## 2nd Law of Motion Explanation

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!

Here’s the breakdown:

• 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.

## Second Law of Motion Formula

We are going to discuss the formula of Newton’s Second Law of Motion. We also call the Second Law of Motion 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

## Second Law of Newton Formula Derivation

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 describes force in a practical way, while Newton’s second law provides a numerical explanation of force.

Consider a body with instantaneous velocity , and momentum given by:

Since, according to the second law of motion,

Where  is the force acting on the object.

Also, since the momentum is defined as,

Therefore, the previous equation becomes,

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,

It is known that the time rate change of velocity of the body is termed as its acceleration i.e.

Therefore,

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

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.

The image given below shows a car of mass ‘M’ accelerating at ‘a’ when the force applied is F.

Newton’ Second law Example

### Newton’s Second Law Formula for Variable Mass

Let us assume a body to be at an initial point (0) specified at location L0 and at the time instance t0. Let us assume the body has mass m0 and travels with a uniform velocity v0. After applying a force of F, the object reaches its destination (point 1) at location L1. This happens at a specific time, t1.

The mass and velocity of the body undergo a transformation as the body travels to v1. Deriving the values for m1 and v1, we get,

Imagine a car at two different points in time as shown in below image. At the first point (marked T0), it’s moving at a certain speed (V0) and has covered a specific distance (L0). Then, at a later point (T1), its speed has changed to V1 and it has traveled a total distance of L1.

### Newton’s Second Law Formula for Constant Mass

For a constant mass, the usage of Newton’s second law can be equated as follows:

### Derivation of Momentum

Momentum is the quantity of motion, which is the product of a body’s mass and velocity. When you walk, run, etc., you have momentum.

The momentum of a body is the product of the mass of the body and its associated velocity. Momentum can be considered to be a vector quantity, that is, it has both an associated magnitude and direction.

Mathematically, the momentum (p) of an object of mass (m) moving with velocity (v) is defined as:

p= m × v

The unit of momentum is kg ms-1 and its dimensional formula is [MLT-1]

The second law of motion gives us a method to calculate the force required to move an object.

## 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.

## Newton’s Second Law of Motion Examples

The second law of motion finds its usage in several domains:

• 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.

Second Law of Motion Example

• A sand bed or cushioned bed is provided to athletes performing long jumps to induce a delay in the change of momentum. The momentum induced because of the velocity and mass of the athlete is reduced to zero as soon as the athlete reaches back on the surface. In case the rate of momentum changes instantly, it may hurt the player. The purpose of the cushioned bed is to delay the momentum of the athlete to zero and thus prevent injury.
• 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.

Newton’s Second Law of Motion Example

• An object falling down from a certain height undergoes an increase in acceleration because of the gravitational force applied.

## Second Law of Motion- Applications

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.

Second Law of Motion Application

• 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.

Resources related to Newton’s second law,

## Second Law of Motion- Solved Examples

Here are some solved examples on Second Law of Motion of Newton :

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?

Solution:

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?

Solution:

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?

Solution:

Given, Mass of truck (m) = 2000 kg

Resistance (F) offered by the ground = – 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]

Formula to calculate the applied force is,

F = m × a

Where m is the mass and a is the acceleration.

⇒ a = F / m

⇒ a = – 4000 N / 2000 kg

⇒ a = -2 m/s2

Now, using the equation of motion

v2 = u2 + 2as

⇒ 0 = u2 + {2 × (-2) × 20}

⇒ u = 80 m/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?

Solution:

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?

Solution:

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?

Solution:

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/s

## Practice Problems on Newton’s Second Law of Motion

We have provided you here with some 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, Force is equal to the rate of change of momentum. This article explored the Second law of motion using its examples, formula, derivation, and applications. Below are some solved numerical questions and practice problems to improve your concept of 2nd law of motion.

## Second Law of Motion- FAQs

### State Newton’s Second Law of Motion.

According to Newton’s second law of motion, “The rate of change of linear momentum is equal to the force applied.” For example the force applied by the brakes of car in stopping it is equal to the rate of change in its momentum.

### Write the formula for Newton’s Second Law of Motion.

According to Newton’s second law of motion the force applied on an object is mass times its acceleration. i.e. F = m×a

### What are some examples of Newton’s Second Law of Motion?

Few examples of Newton’s second law of motion include,

• Pushing harder to increase the speed of your bicycle.
• Accelerating to increase the speed of a car.
• Kicking a football to increase its speed.

### What is the other name for Newton’s Second Law of Motion?

As Newton’s Second Law gives the quantitative description of the force, thus Newton’s Second Law of Motion is also known as the quantitative law of motion and real law of motion.

### Which Law is the Special Case of Newton’s Second Law of Motion?

A special case of Newton’s second law of motion is Newton’s first law of motion.

### How does the Second law of motion by Newton apply to rockets?

Newton’s second law governs how rockets work. The rocket’s speed-up (acceleration) depends on how much power it blasts out (force). The stronger the blast off, the faster the rocket goes! The rocket’s weight (mass) also matters. A heavier rocket needs a larger push to reach the same speed. The law helps us understand how rockets can move and change velocity in space.

### How does Newton’s second law apply to a car crash?

The force of impact experienced in a car crash, denoted as F, is directly proportional to both the car’s mass (m) and the rate of change in its velocity (acceleration, a). In simpler terms, the severity of a crash is directly related to the car’s speed and weight. A greater mass or velocity results in a proportionally higher force of impact.

### How is newton’s second law of motion real law of motion?

Newton’s second law of motion contains within itself the first and third laws of Newtonian mechanics, hence it is the real law of motion.

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