   Momentum and Second Law of Motion

# Second Law of Motion

The first law of motion indicates that when an unbalanced external force acts on an object, its velocity changes, that is, the object gets an acceleration. We would now like to study how the acceleration of an object depends on the force applied to it and how we measure a force. Let us recount some observations from our everyday life. During the game of table tennis if the ball hits a player it does not hurt him. On the other hand, when a fast moving cricket ball hits a spectator, it may hurt him. A truck at rest does not require any attention when parked along a roadside. But a moving truck, even at speeds as low as 5 m s-1, may kill a person standing in its path. A small mass, such as a bullet may kill a person when fired from a gun. These observations suggest that the impact produced by the objects depends on their mass and velocity. Similarly, if an object is to be accelerated, we know that a greater force is required to give a greater velocity.

In other words, there appears to exist some quantity of importance that combines the object's mass and its velocity. One such property called momentum was introduced by Newton.

### Let us now define momentum..

The momentum, p of an object is defined as the product of its mass, m and velocity, v. That is,

P=mv  ----------- (1)

Momentum has both direction and magnitude. Its direction is the same as that of velocity, v. The SI unit of momentum is kilogram-metre per second (kg ms-1). Since the application of an unbalanced force brings a change in the velocity of the object, it is therefore clear that a force also produces a change of momentum.

## Mathematical Formulation of Second Law of Motion

Suppose an object of mass, m is moving along a straight line with an initial velocity, u. It is uniformly accelerated to velocity, v in time, t by the application of a constant force, F throughout the time, t.

The initial and final momentum of the object will be, p1 = mu and p2 = mv respectively.

The change in momentum ∝ p2 – p1
∝ mv – mu
∝ m × (v – u).
The rate of change of momentum ∝ m × (v −u) / t

Or, the applied force,
F ∝m × (v −u) / t

Or, the applied force,  F  =  km × (ν −u) / t           ----------- (2)
= kma                           ------------(3)

Here  [a = (v – u) / t ] is the acceleration, which is the rate of change of velocity. The quantity, k is a constant of proportionality. The SI units of mass and acceleration are kg and m s-2 respectively. The unit of force is so chosen that the value of the constant, k becomes one. For this, one unit of force is defined as the amount that produces an acceleration of 1 ms-2 in an object of 1 kg mass. That is,

1 unit of force = k × (1 kg) × (1 m s-2)

Thus, the value of k becomes 1. From Eq. (3) , therefore, F = ma  ---------(4)

The unit of force is kg m s-2 or newton, which has the symbol N. The second law of motion gives us a method to measure the force acting on an object as a product of its mass and acceleration.

The second law of motion is often seen in action in our everyday life. Have you noticed that while catching a fast moving cricket ball, a fielder in the ground gradually pulls his hands backwards with the moving ball? In doing so, the fielder increases the time during which the high velocity of the moving ball decreases to zero. Thus, the acceleration of the ball is decreased and therefore the impact of catching the fast moving ball is also reduced. If the ball is stopped suddenly then its high velocity decreases to zero in a very short interval of time. Thus, the rate of change of momentum of the ball will be large. Therefore, a large force would have to be applied for holding the catch that may hurt the palm of the fielder. In a high jump athletic event, the athletes are made to fall either on a cushioned bed or on a sand bed. This is to increase the time of the athlete ís fall to stop after making the jump. This decreases the rate of change of momentum and hence the force. Try to ponder how a karate player breaks a slab of ice with a single blow.

The first law of motion can be mathematically stated from the mathematical expression for the second law of motion. Eq

F = ma

or   F= m (v- u) / t  --------------(5)
or Ft = mv – mu

That is, when F = 0, v = u for whatever time, t is taken. This means that the object will continue moving with uniform velocity, u throughout the time, t. If u is zero then v will also be zero. That is, the object will remain at rest.

# Lesson Summary:

1. Second law of motion: The rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of the force.

2. The SI unit of force is kg m s–2. This is also known as Newton and represented by the symbol N. A force of one Newton produces an acceleration of 1 ms–2 on an object of mass 1 kg.

3. The momentum of an object is the product of its mass and velocity and has the same direction as that of the velocity. Its SI unit is kg m s–1.

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