Question

Derive the relationship F = ma, where each symbol has its usual significance.

Answer 1

Suppose an object of mass, m is moving along a straight line with an initial velocity, u. It is uniformly accelerated to velocity, v in time, t by the application of a constant force, F throughout the time, t. The initial and final momentum of the object will be, $p_1=mu$ and $p_2=mv$ respectively.

$Thechangeinmomentum\propto p_2–p_1$

$\propto mv–mu$

$\propto m\times (v-u)$
The rate of change of momentum $\propto \frac{m(v-u)}{t}$
Or, the applied force,
$F\propto \frac{m(v-u)}{t}$
$F=\frac{km(v-u)}{t}$
$F=kma$
Here $[a=\frac{(v-u)}{t}]$ is the acceleration, which is the rate of change of velocity. The quantity, k is a constant of proportionality. The SI units of mass and acceleration are kg and $m{s}^{-2}$ respectively. The unit of force is so choosen that the value of the constant, k becomes one. For this, one unit of force is defined as the amount that produced an acceleration of $1m{s}^{-2}$ in an object of 1 kg mass. That is, $1unitofforce=k\times \left(1kg\right)\times \left(1m{s}^{-2}\right)$.
Thus, the value of k becomes 1.Hence
F=ma
The unit of force is $kgm{s}^{-2}$ or newton, which has the symbol N. The second law of motion gives us a method to measure the force acting on an object as a product of its mass and acceleration.