Question

explain newtons second law of motion mathematicly​

Answer 1

Answer:

the rate of change of momentum of a body directly proportional to the applied force and take place in the direction in which the force acts

1. F oo dp/dt
2. F =k×dp/dt

Answer 2

፨ Required Answer :

According to Newton's 2nd law, the change in momentum of a body per unit time is directly proportional to the unbalanced force acting on the body and the change in momentum takes place in the direction of the unbalanced force on the body.

That is, F is directly proportional to dp/dt

Where,

• dp is the Change in momentum,
• dt is the time taken for this change in momentum

MATHEMATICAL FORMULATION OF SECOND LAW OF MOTION

Consider a body of mass m moving with initial velocity u. Let a force F acts on the body for time t so that the velocity of the body after time t is v.

Initial momentum = mu

Final momentum = mv

Now ,

Change in momentum :

→ mv - mu

→ m(v - u)

Time taken to change this momentum :

→ (t - 0) = t

Therefore ,

Rate of change of momentum = Change in momentum/Time taken

Rate of change of momentum = m(v - u)/t

According to the definition of Newton's second law of motion, $\:$

Force applied is directly proportional to rate of change of momentum

Or ,

F is directly proportional to

→ m(v - u)/t.................❶

Since v = u + at or (v - u/t) = a

Therefore, eqn. ❶ can be written as

F is directly proportional to ma

Or ,

F = kma...........❷

Where,

• k is constant of proportionality

If F = 1 unit

m = 1 unit

And, a = 1 unit

Then from eqn ❷,

→ 1 = k or k = 1

Put this value of k = 1 in eqn ❷, we get,

→ F = ma

Thus, force acting on the body is directly proportional to

1. It's mass
2. It's acceleration.

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