Mathematical Formulation Of Second Law Of Motion

We often observe that, if same magnitude of force is used to push two blocks of wood, where one of the blocks is heavier than the other, the rate of change of position of the lighter block will be more than the heavier one. Similarly, when two same forces are applied to push a car and a bus, the car will have more acceleration compared to the bus. From these examples, it is clear that the acceleration gained by an object when subjected to the same magnitude of force, is a factor of the mass of the object.

Similarly, a small bullet when at rest doesn’t harm the person handling it whereas when loaded in a gun and fired with high velocity, it can kill a person. Hence, it can be concluded that along with mass, velocity also affects the impact produced by any object. Taking another example, when a car is given a momentary jerk, it may not move from its initial position whereas when an extended and continuous force of the same magnitude is applied on the car, it experiences a displacement. With these examples, we can conclude that the impact produced by an object depends on its mass and velocity i.e., its momentum and the time rate at which the change in momentum is occurring. The second law of motion is used to validate this phenomenon. In this section we shall learn about the formulation of the second law of motion.

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Newton’s Second Law of Motion

Newton’s second law of motion can be formally stated as follows: 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.

Mathematical Formulation of Second Law of Motion

Let us consider an object of mass m, moving along a straight line with an initial velocity u. Let us say, after a certain time t, with a constant acceleration, the final velocity becomes v.

Here we see that, the initial momentum

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The final momentum

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The change in momentum can be written as

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As we know, the rate of change of momentum with respect to time is proportional to the applied force.

The applied force

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Or,

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as acceleration (a) = rate of change of velocity with respect to time.

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where k = constant of proportionality.

The SI units of mass and acceleration are kg and m s-2 respectively.

So,

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The second law of motion gives us a method to measure the force acting on an object as a product of mass of the object and the acceleration of the object which is the change in velocity with respect to time.


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