Question

Derive f=ma by calculus method

Answer 1

F is directly proportional to mv-mu/t

F is directly proportional to m(v-u) /t

F is directly proportional to ma......{(v-u) /t}

F = kam......(k = constant)

hence F = ma

hope it's helpful.....

Answer 2

Force = Mass x Acceleration

According to newton's second law of motion, The force acting on the body is directly proportional to law of conservation of linear momentum of the body and the change in momentum takes place in the direction of force.

Let m be the mass of the moving body moving along the straight line, with an initial speed u.

After the time interval of t, the velocity of the body changes to v due to the impact of unbalanced external force f.

Initial momentum of the body Pi = mu

Final momentum of the body Pf = mv

Change in momentum Delta P = Pf - Pi

= mv - mu

By Newton's second law of motion,

Force , F ∝ Rate of change of momentum

$F ∝ \frac{change \: in \: momentum}{time}$

$F∝ \frac{mv - mu}{t}$

$F∝ \frac{km(v - u)}{t}$

Here, k is the proportionality constant, k = 1 in all system of units

$F = \frac{m(v - u)}{t}$

Since,

$acceleration = \frac{change \: in \: velocity}{time}$

$a = \frac{(v - u)}{t}$

Hence we have,

$F = m \times a$