Noninertial Frame And Fictitious Forces
Noninertial frame and Fictitious forces: A football is kept in a stationary bus and we can find that it would at rest. If the bus starts moving, we will find the ball moving from its initial position without applying any external force by the way of interaction with other objects. The above movement is as a result of accelerating frame (bus). Any frame that is found to accelerate is called non inertial frame. Here, Newton's second law 'f = ma ' cannot be used without accounting for additional force acting on the object due to accelerated frame. We turn our attention to appearance of physical laws to an observer in a system accelerating at rate with respect to, inertial system. By rewriting the acceleration relation,
where is the acceleration of primed coordinate system. By multiplying the mass of the body, 'm' we get
or
Here is the true force due to the physical interaction, is the force due to physical interactions observed in the primed coordinate system (apparent force) and is the fictitious force. Fictitious force in some text is called inertial force. The fictitious force experienced in a uniformly accelerating system is uniform and proportional to the mass of the system, like a gravitational force. The fictitious forces originate from the acceleration of coordinate system and not due to interaction between bodies.
Galilean Transformation and Velocity of Light: The Galilean transformation successfully explained the invariance of laws of Newtonian mechanics in different inertial frame. No experiment in physics carried out in a single inertial frame can tell us that our frame is at rest or moving with uniform velocity. Let me remind you that our earth moves with a velocity of 30 Km/sec with respect to space in a nearly circular path around the sun. There is no preferred inertial frame for the laws of mechanics to hold true.
It was believed that light travels with above velocity in a medium called ether, which is inertial with respect to sun.
According to Galilean transformation (v'=vV), for an observer in a S'frame moving with velocity V would measure the velocity of light pulse ranging from cV to c V depending upon the relative direction of c and V. But according to Maxwell's equation, the velocity of light is constant in empty space/vacuum. The consequence of above observation has following possibilities:

If both Galilean and Maxwell's equations are correct, then there is a preferred inertial frame for laws of mechanics and electromagnetic theory.

If Maxwell's equations are incorrect, we should be able to prove with the help of some experiments.

If Galilean transformations are incorrect, we have to reformulate Newtonian mechanics also, because Newtonian mechanics is explained by Galilean transformation.
Example 1 : A man traveling in a train walks with a speed of 6 km/h with respect to train and opposite to the direction of movement of train. The velocity of the train with respect to ground is 80 km/h. What is the velocity of man observed by an observer on the ground?
Solution: Here ground is the rest frame 
Train is the internal frame 
According to Galilean transformation,
Here,
Example 2 : A small lantern of mass 100 gm is hanged from the pole of a bullock cart using a massless string. What is the tension in the string,
(a) when the bullock cart is at rest,
(b) moves with uniform velocity of 3 m/s
(c) moves with an acceleration of 2 . Assume that cart moves smoothly.
Solution:
(b) Uniform Velocity (Inertial frame) :No acceleration acts on the system. So.
Here also