Question

Describe Newton's law of motion giving examples and also derive the relationship between force, mass, and acceleration of a body based on the second law.

Answer 1

Hint: Newton's laws of motion are the basis of all the motion happening in the universe.
There are three laws of motion:
First law - An object at rest stays will remain at rest and an object in motion will stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
Second Law - The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
Third law- Every action has an equal opposite reaction

Complete solution:
Newton's first law of motion is often stated as: An object at rest stays at rest and an object in motion will stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
Or we can say that the position of a body can only be changed by applying an external force
Newton's first law of motion is also called the law of inertia
Inertia: a property of matter by which it remains at rest or in uniform motion in the same straight line unless acted upon by some external force
Example
1. A ball kept on the floor when you kick it moves through a distance.
Explanation: In this case, the initial ball was at rest and when you kicked the ball you applied force on it which changed its state, and the ball moved.
2. A ball rolling on the ground stops after some distance due to frictional force offered by the rolling surface.
Newton’s Second law of motion is stated as the rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of the motion
Momentum can be defined as "mass in motion." All objects have mass; so if an object is moving, then it has momentum - it has its mass in motion. Momentum depends upon the variables mass and velocity. In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object.
Momentum$\left( \rho \right) = $mass $ \times $acceleration
$\rho = m \times v$
Now, According to Second law,
Applied force $ \propto $rate of change of momentum
Let us suppose there is a body moving with an initial velocity$u\dfrac{m}{s}$, a force f is applied on it which changes its velocity to\[\;v\]$\dfrac{m}{s}$
Now, As
$ f \propto \dfrac{{\Delta \rho }}{t}$
$\Delta \rho = mv - mu$
$\Rightarrow \Delta \rho = m\left( {v - u} \right)$
Rate of change of momentum $\left( {\dfrac{{\Delta \rho }}{t}} \right) = m\dfrac{{\left( {v - u} \right)}}{t}$
Change of velocity with time is called acceleration
$\therefore \dfrac{{\left( {v - u} \right)}}{t} = a$
On Substituting we get
$\left( {\dfrac{{\Delta \rho }}{t}} \right) = m \times a$
$\therefore f \propto m \times a$
To remove the proportionality we introduce `k` as a proportionality constant
$\therefore f = k \times m \times a$
Now, let us suppose
If the mass of the body is 1kg and it is moving with acceleration on 1$\dfrac{m}{s}$, the 1 unit of force is given by
1 unit force $ = k \times 1kg \times 1\dfrac{m}{s}$
$\therefore$ The value of k comes out to be 1.
The final result comes out to be $f = m \times a$.
Example: a car is moving on a straight road with a uniform velocity, another car comes from behind and hits it due to this collision. The second car applied force on the first car and gave acceleration to it.
Third law of motion: The third law states that for every action (force) in nature there is an equal and opposite reaction. In other words, if object A exerts a force on object B, then object B also exerts an equal and opposite force on object A.
Example:
1. When you slap someone you also feel pain in your palms. This is because when you apply force by slapping someone the reaction force of equal magnitude is exerted bask on your hand
2. When you step out of a boat as soon as you move out the boat is pushed back into the river, this is also due to the force applied by you on the boat when you move out of the boat due to its reactionary force boat moving away from you.

From the second law of motion it is concluded force applied on an object is the product of its mass and acceleration.

Note: 1. Rockets traveling through space encompass all three of Newton's laws of motion.
2. The larger the mass the more will be applied to the object.
3. If there will be no friction the rolling body will never stop on its friction offers resistance to the motion of the ball and finally stops it.
4. Momentum is always conserved.