Question

State and explain Newton’s second law of motion. Derive the expression F=ma.

Answer 1

Hint: According to Newton’s second law of motion the force is equal to the rate of momentum. The change in momentum which is also known as impulse is the product of the mass and the velocity. Then the force is the product of mass and acceleration. Hence the unit of force is Newton or $\dfrac{kg}{{{m}^{2}}}$.

Formula used:
$F=\dfrac{dp}{dt}$
where, F is the force
$\dfrac{dp}{dt}$ is the rate of change of momentum.

Complete answer:
Initial momentum, ${{p}_{i}}=mu$
Final momentum, ${{p}_{f}}=mv$
Change in momentum, $\vartriangle p={{p}_{f}}-{{p}_{i}}$
$\begin{align}
  & \Rightarrow \vartriangle p=mv-mu \\
 & \vartriangle p=m\left( v-u \right) \\
\end{align}$
Time taken to cause change in momentum$=\vartriangle t$
$\therefore $ Rate of change of momentum$=\dfrac{\vartriangle p}{\vartriangle t}$ $=\dfrac{m\left( v-u \right)}{\vartriangle t}$
F$=m\left[ \dfrac{\vartriangle v}{\vartriangle t} \right]$
F$=ma$
According to Newton’s second law of motion force is proportional to the acceleration. Or otherwise, change in momentum is equal to impulse. Where impulse is the product of force and time. For a rigid body the total momentum of the body will be the sum of individual moments of particles. Since, velocity is a vector quantity the momentum is also a vector quantity. That is, which have both magnitude as well as direction. The momentum has the same direction as velocity since it directly depends on velocity. Momentum can also be described as a type of speed or force of a moving object. Whereas, the inertia is the tendency of an object to move on. Thus all the moving objects will have a momentum. The momentum of a body depends upon its mass and velocity. Hence the body which has higher mass or velocity will have greater value of momentum. Thus momentum of a particle may be explained as the product of its mass and velocity.

Note:
Thus force is equal to the rate of change of momentum. According to Newton’s second law of motion the force is equal to the rate of change of momentum. The change in momentum which is also known as impulse is the product of the mass and the velocity. The momentum has the same direction as velocity since it directly depends on velocity. Momentum can also be described as a type of speed or force of a moving object.