As action and reaction forces are equal and opposite why do not they cancel each other
Newton's third law of motion, which states that for every action force there is an opposed and equal reaction force. What this means is that when one object exerts a force on another object, that second object exerts an equal magnitude and opposite-direction force back on the first.
It doesn't really matter which force you call the action or reaction. What's most important to understand is that they go together and these two forces are always equal in strength and opposite in direction.
It's fairly simple to identify the action and reaction forces between objects. First, it helps to identify the interaction itself. Let's say you are doing a push-up the floor. Or, maybe a tennis racket is hitting a tennis ball. How about a rocket launching off the ground?
All of these examples are interactions that result in forces being exerted on the objects involved in the interaction. When you push against the floor, you exert an action force onto the ground, while the ground exerts a reaction force right back on you.
In a tennis match, the racket exerts the action force on the ball and, as the ball hits it, it exerts an equal and opposite reaction force on the racket.
The rocket launches because it pushes on the gas coming out the back end for the action force, while the gas pushes the rocket upward with a reaction force.
In each of these cases, the action is the force exerted by the first object on the second, and the reaction force is the force exerted on the first object by the second. The important thing to notice is that action and reaction forces act on different objects in the interaction. Two equal and opposite forces acting on the same object do not make an action-reaction pair!
This is why the two opposing forces don't cancel out. If a football is kicked from both sides at the same time, it doesn't go anywhere, because the two feet kicking it exert equal and opposite forces on the same ball and cancel each other out.
However, when you kick a football with just one foot, that foot exerts an action force on the ball and, at the same time, the ball exerts a reaction force on your foot. In this case, the forces don't cancel out, so the ball travels through the air.
You may be wondering why the ball takes off but you stay standing on the ground. This has to do with the mass of the objects in the interaction. You are far more massive than the ball, so the ball accelerates. Both objects exert the same amount of force, but what's different is the effect of the forces on the different objects.
Let's take a closer look at this. Think of a cannon firing a cannonball. There is an interaction between the two objects, so there are action and reaction forces. However, the cannon recoils only slightly as it shoots out the cannonball, but the cannonball goes flying through the air.
We can look at Newton's second law of motion to see why this happens. This law states that acceleration is proportional to the net force and inversely proportional to the mass of the object. What this means is that when the force increases, so does the acceleration, which is a change in an object's state of motion. But as the mass increases, the acceleration decreases. Since the forces are equal, you can see that mass is really the key player in how an object is affected by the forces acting on it.
Consider another example...
You push a trolley full of....books; you move it (effect) so that it acquires an acceleration (it was at rest and now it moves).
The trolley exerts on your hand a reaction that you can detect by feeling an indent into your skin. Now increase the number of books inside the trolley; the movement (to start it, I mean) becomes each time more difficult...up to a point where you'll not be able to move the trolley.
The mystery here is MASS!
This is the property of matter to RESIST to the action of the force and in doing so modify the ACCELERATION produced.
In simple words:
the EFFECT =a=F/m
The bigger the mass the smaller will be the acceleration!!!
In our case it is true that the two forces are equal but at the start your mass is far bigger than the mass of the trolley+books so the acceleration on you is negligible and masked by friction (the sole of your feet) while on the trolley is big (you can magnify this effect on you by passing on a spot covered with lubricant oil while pushing the trolley....now you'll "feel" the reaction by slipping...this is the trolley "pushing" back at you!).
When the two masses (you and trolley) are the same the effects (accelerations) will be both eliminated indeed!