Question

State Newton’s second law of motion and hence derive the relation F = ma.​

Answer 1

Newton's Second Law of Motion :

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

⇢ Force ∝ Change in momentum/ Time taken

⇢ F ∝mv - mu/t

⇢ F ∝ m (v - u)/t

• v - u/t = a

⇢ F ∝ ma

⇢ F = k × m × a

Where, k is constant

⇢ F = ma

⇢ Force = Mass × Acceleration

_____________________________

More details :-

First law of motion :

A body at rest will remain at rest,and a body in motion will continue in motion in a straight line with uniform speed, unless it is compelled by an external force to change its state of rest or of uniform motion.

Third law :-

To every action there is an equal and opposite reaction.

Answer 2

Answer :-

Statement:-

★ Force acting on an object is directly proportional to the rate of change of its momentum with respect to time. It takes place in the direction in which the force acts.

Why we need to prove everything !

Anyways let's start ^^"

Here we go ...

Derivation:-

Derivation by Calculus Method:-

Consider a body of mass m moving with a velocity of u

f force act on the body for time T velocity after t is v

Then ,

Initial momentum of body is mu

Final momentum of body is mv

Change in momentum is ( mv - mu )

↠ m{ v - u }

Time taken = t - 0 ↠ t

Rate of change of p ↠ Change in momentum / Time taken

↠ m( v - u ) / t. ......... 1

According to statement the force is directly proportional to the rate of change of momentum. So,

↠F proportional to m( v - u ) / t

From 1st equation of motion

↠ v = u + at

OR

↠ v - u / t = a .......... 2

From 1 & 2

F proportional ma

OR

↠ F = kma. { where k is constant }

F = kdp/dt (p= momentum)

F = dp/dt (k= constant=1)

F = dmv/dt (p=mv)

F = mdv/dt

Unit:-

↠Newton or kgm/sec²

↠ Dyne(CGS unit)

↠ $1 \:N$ = 10^5 dyne

Newton’s second law of motion states :-

❥ 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.

❥ the first one is done using differential process.

Newtons law uses :-

❥ Using Newton's Second Law we can know how much force is acting on the car.

❥ If the forces are unbalanced, the object will accelerate.

❥ F = ma is very important because it shows the relationship between forces and motion.

❥ It allows you to calculate the acceleration.

Examples :-

↠ Cricket player increases time to reduce force or momentum.

↠ Karate player reduce time to increase force.