Question

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

Answer 1

Answer:

Newton's second law of motion states that the acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system, and inversely proportional to its mass. In equation form, Newton's second law of motion is a=Fnetm a = F net m .

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,

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

In scalar form, this equation can be written as F = ma.

• The Force is a vector quantity.