Question

Using second law of motion derive the relation between force and acceleration

Answer 1

Answer:

f=ma

Explanation:

According to second law of motion Force is directly proportional to rate of change of momentum.

so F is directly proportional to $\frac{dp}{dt}$

we know that momentum P= MV

so

$f=d(mv)/dt=m\frac{dv}{dt}$

thus we get F=ma.

Answer 2

Answer:

Let F be external force applied on the body in the direction of motion of the body for time interval $\sf \Delta t$, the the velocity of a body of mass m changes from $\sf v$ to $\sf v + \Delta v$ i.e. change in momentum, $\sf \Delta p = m \Delta v$.

According to Newton's second law :

$:\implies \sf F \propto \dfrac{\Delta p}{\Delta t}$

$:\implies \sf F = k \: \dfrac{\Delta p}{\Delta t}$

Where k is a constant of proportionality.

If limit $\sf \Delta t \rightarrow 0,$ then the term $\sf \dfrac{\Delta p}{\Delta t}$ becomes the derivative $\sf \dfrac{dp}{dt}$.

Thus,

$:\implies \sf F = k \: \dfrac{dp}{dt}$

For a body of fixed mass (m), we have :

$:\implies \sf F = k \dfrac{d(mv)}{dt}$

$:\implies \sf F = km \: \dfrac{dv}{dt}$

$:\implies \sf F = kma$

If v is fixed and m is variable then :

$:\implies \sf F = \dfrac{kd(mv)}{dt}$

$:\implies \sf F = \dfrac{kvdm}{dt}$

because, k = 1 then :

$:\implies \sf F =\dfrac{vdm}{dt}$

Now, a unit force may be defined as the force which produces unit acceleration in a body of unit mass :]

So,

F = 1

m = 1

a = 1

k = 1

So,

$:\implies \underline{ \boxed{ \sf F = ma}}$