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# Newton’s Second Law Of Motion: Definition, Examples, Derivation of First Law Using Second Law, Impulse

Newton’s Second Law Of Motion , First Law Using Second Law, Impulse, Practice Problems, FAQs

If we release the tennis ball and paper ball from the top of the building, which one will reach first on the ground? You might say tennis ball because it’s heavier. But in an ideal situation both will reach the ground simultaneously. The gravitational force acting on both the objects is different but they have the same acceleration downwards. This relation between force, mass and acceleration is given by the second law of motion. Let’s learn about this law in detail.

• Newton’s second law of motion
• Mathematical formulation of second law
• First law contained in second law
• Impulse
• Practice problems
• FAQs

## Newton's Second Law of Motion

Newton’s second law of motion, unlike the first law of motion, pertains to the behaviour of objects for which all existing forces are unbalanced. The second law is more quantitative in natureand is used to analyse situations involving a force.

Newton’s second law of motion states that the rate of change of momentum of a body in a given inertial reference frame is directly proportional to the applied force and it takes place in the direction in which the force acts.

Note: Second law is also called the law of momentum.

Example: Man is trying to move two boxes of mass and () with the same acceleration.

According to Newton’s second law, higher mass will require higher force to move. So he will have to apply higher force to move block than . This applying of force changes the state of the block from rest.

Example2:

Vary the force by keeping the mass of the block constant. Measure the average acceleration for the same time interval. It is observed that the force is directly proportional to the average acceleration measured. It can be plotted as shown in the figure below.

Conclusion

## Mathematical Formulation of Second Law

Let’s derive the mathematical equation of Newton's second law.

Consider a body of mass moving with some initial velocity if unbalanced net force is applied velocity will change from to assuming mass to be constant.

Therefore momentum changes from (Initial) to (final)

Change in momentum will be

According to second law,

. . . . . is proportionality constant

Taking the limit , the term become the derivative or differential coefficient of with respect to time denoted by .

We know,

…….considering constant mass

………..()

By using experiments value of is found to be 1,

Here,

## First law contained in second law

Let’s see how the first law can be deduced from Newton’s second law of motion.

We know,

We can write it as,

…….. is the final velocity and is the initial velocity of an object.

If the net force on an object is zero,

Then,

We can conclude that, if

• Since object will continue moving with uniform velocity .
• If then That means the object will remain at rest.

Therefore, we can say that Newton’s first law is contained in the second law.

## Impulse

You must have observed that while catching a cricket ball a player moves his hands backwards. Cricket ball coming towards a fielder has very large momentum. By moving hands backward, what does the player want to achieve ? Have you ever thought about it ? That’s where the concept of Impulse comes into play.

• Impulse is defined as the change in momentum. It is measured as the product of average force and time for which force acts.
• When a large force works on a body for a very short time interval, it is called an impulsive force.
• An impulsive force is not constant, in fact it changes its value from zero to maximum then from maximum to zero.
• It is a vector quantity directed along the direction of force.
• It is denoted by .
• ……..here is momentum

From Newton's second law we know,

We can write it as,

Integrating both sides,

Therefore for a small interval of time, Impulse is written as,

Note: Area under Force vs Time graph gives Impulse.

Let’s revisit the example we have taken at the start. Now we can understand what the player is trying to achieve by moving his/her hands backwards. By doing so he increases the time interval to reduce the momentum of the ball. Rate of change of momentum becomes slow. So less force is required to stop the ball. So his/her hands are saved from getting hurt.

From equation as increases, decreases.

Video Explanation: Second law of motion

## Practice Problems

Q1. A force acts on an object moving in the xy plane. Find the magnitude of impulse on an object in the time interval to .

Solution:

Impulse,

Q2. An object of mass is subjected to two forces and . Then find the magnitude of acceleration?

Solution:

Q3. The momentum of a particle moving in straight line is given as . Find the force on a particle at ?

Solution: from newton's second law,

At t=5 sec

Q4. A force-time graph for linear motion is shown in figure. Find the change in linear momentum from ?

Solution: Area under force-time graph gives the change in momentum of the body during that interval.

From the given graph, Area under the curve is,

## FAQ

Q1. Can the first law of motion be derived from Newton’s second law of motion?
Solution: Yes, absolutely. Newton’s first law of motion can be derived from the second law.

Q2. Does the value of force depend on the frame of reference?
Solution: No. Force does not depend on the frame of reference.

Q3. What is the relation between and ?
Solution:

Q4. What does the area under force vs time graph give?
Solution: Area under force-time graph gives the change in momentum in that interval.

## NCERT Class 11 Physics Chapters

 Physical World Units and Measurements Motion in a Straight Line Motion in a Plane Laws of Motion Work Energy and Power Particles and Rotational Motion Gravitation Mechanical Properties of Solids Mechanical Properties in Liquids Thermal Properties of Matter Thermodynamics Kinetic Theory Oscillations Waves
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