Question

Derive mathematical relestion of newton 2nd law of motion considered and objects of mass m moving allong a straight line with an inertia

Answer 1

To do:

• Deriving the mathematical relation of newton's second law of motion. taking object of mass 'm' moving along a straight line with an inertia.

Solution:

According to the Newton's second law of motion:

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

So,

If a force F is acting on a body of constant mass m, for a time interval of Δt, to change the velocity of body, v by a value Δ v.

Then, Initial momentum of body p = m v will change by Δp = m Δv.

Now, According to the second law of motion

→ F ∝ Δp / Δt

→ F = k (Δp / Δt)

[ where k is proportionality constant ]

Now, taking limit Δt → 0 we will get,

→ F = k (dp / dt)

so,

→ F = k [d(mv) / dt]

→ F = k m ( dv / dt )

∵ [ dv / dt = a ]

→ F = k m a

so, we have Newton's second law as

→ F = k m a

that means Force is proportional to the product of mass and acceleration.

so, Now taking k = 1

→ F = ( 1 ) m a

→ F = m a

Here, Mathematical relation of newton's second law of motion is derived.

Answer 2

Explanation:

→ Newton's Second law of motion :

✮ Newton' s Second law

The force acting on a body is directly proportional to the rate of change of linear momentum of a body and the change in momentum takes place in the direction of the force.

Let m be the mass of the moving body along a straight line with initial velocity (u).After some time (t) The velocity becomes (v) due to the unbalanced force F.

Initial momentum of the body = mu

Final momentum of the body = mv

Change in momentum ∆P

= mv-mu = m ( v - u )

Force is directly proportional to the change in momentum.

F = m ( v - u ) / t

Since Acceleration = change in velocity/time taken

a = ( v-u) / t

Therefore F = ma

Force = mass × acceleration

Thus we arrived the Newton's Second Law of motion

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