Momentum and Second Law of Motion

If a cricketer catches a ball he moves his hand back while catching the ball. He does this to reduce the impact, due to the force of the ball on his hand. An object in motion has momentum. Momentum is defined as the product of mass and velocity of an object.

The momentum of the object at the starting of the time interval is called the initial momentum and the momentum of the object at the end of the time interval is called the final momentum. The rate of change of momentum of an object is directly proportional to the applied force.

Newton's second law quantifies the force on an object. The magnitude of force is given by the equation,
F = ma, where 'm' is the mass of the object and 'a' is its acceleration. The CGS unit of force is dyne and the SI unit is newton (N).

A large amount of force acting on an object for a short interval of time is called impulse or impulsive force. Numerically impulse is the product of force and time. Impulse of an object is equal to the change in momentum of the object.

Impulse and Impulsive Force
The momentum of an object is the product of its mass and velocity. The force acting on a body causes a change in its momentum. In fact, according to Newton’s second law of motion, the rate of change in the momentum of a body is equal to the net external force acting on it.

Another useful quantity that we come across is “impulse”. “Impulse” is the product of the net external force acting on a body and the time for which the force is acted.

If a force “F” acts on a body for “t” seconds, then Impulse I = Ft.

In fact, this is also equal to the change in the momentum of the body. It means that due to the application of force, if the momentum of a body changes from “P” to “P ' ”, then impulse,
I =  P ' - P.

For the same change in momentum, a small force can be made to act for a long period of time, or a large force can be made to act for a short period of time. A fielder in a cricket match uses the first method while catching the ball. He pulls his hand down along with the ball to decrease the impact of the ball on his hands.



In a cricket match, when a batsman hits a ball for a six, he applies a large force on the ball for a very short duration. Such large forces acting for a short time and producing a definite change in momentum are called “impulsive forces”.

Derivation of Newton’s Second Law of Motion
Newton’s second law of motion states that the rate of change of momentum of an object is Proportional to the applied unbalanced force in the direction of force.

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
The change in momentum = mv – mu
The change in momentum = m × (v – u).
The rate of change of momentum = m × (v - u) t
 
Or, the applied force,

 F ∝ m × (v - u) t    
 F = km × (v - u) t   = kma     ---------------------------- (i)
 
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 m s-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. (i)

F = ma 
The unit of force is kg m s-2 or newton, represented as N.

Derivation of Newton’s first law of motion from Newton’s Second Law of Motion
Newton's first law states that a body stays at rest if it is at rest and moves with a constant velocity if already moving, until a net force is applied to it. In other words, the state of motion of a body changes only on application of a net non-zero force.

Newton's second law states that the net force applied on a body is equal to the rate of change in  its momentum. Mathematically,
 
      F d p dt    
 
Where, F is the net force, and p is the momentum. Now, we can write the same as:
 F =   d p dt d( mv ) dt
 ⇒  F  = d(m v ) dt    
⇒   F → = md v dt    
                
So, if the net force, F is zero, change in the value of v is be zero i.e., a body at rest will be at rest and a body moving with constant velocity will continue with the same velocity, until a net force is applied. This conclusion is similar to the Newton's first law of motion.
 
Thus, we can derive Newton’s first law of motion using Newton’s second law of motion.
 

Derivation of Newton’s third law of motion from Newton’s second law of motion
Consider an isolated system of two bodies A & B mutually interacting with each other, provided there is no external force acting on the system. 

Let FAB, be the force exerted on body B by body A and FBA be the force exerted by body B on A.
 
Suppose that due to these forces FAB and FBA, dp1/dt and dp2/dt be the rate of the change of momentum of these bodies respectively. 

Then, FBA dp1 dt     ---------- (i) 
     => FAB =   dp2 dt     ---------- (ii) 

Adding equations (i) and (ii), we get, 
FBA + FAB =   dp1 dt +  dp2 dt  

⇒ FBA + FAB = d(p1 + dp2) dt     

If  no external force acts on the system, then 

d(p1 dp2)dt = 0

⇒ FBA + FAB = 0
⇒ FBA = - FAB---------- (iii) 

the above equation (iii) represents the Newton's third law of motion (i.e., for every action there is equal and opposite reaction)...

Summary

If a cricketer catches a ball he moves his hand back while catching the ball. He does this to reduce the impact, due to the force of the ball on his hand. An object in motion has momentum. Momentum is defined as the product of mass and velocity of an object.

The momentum of the object at the starting of the time interval is called the initial momentum and the momentum of the object at the end of the time interval is called the final momentum. The rate of change of momentum of an object is directly proportional to the applied force.

Newton's second law quantifies the force on an object. The magnitude of force is given by the equation,
F = ma, where 'm' is the mass of the object and 'a' is its acceleration. The CGS unit of force is dyne and the SI unit is newton (N).

A large amount of force acting on an object for a short interval of time is called impulse or impulsive force. Numerically impulse is the product of force and time. Impulse of an object is equal to the change in momentum of the object.

Impulse and Impulsive Force
The momentum of an object is the product of its mass and velocity. The force acting on a body causes a change in its momentum. In fact, according to Newton’s second law of motion, the rate of change in the momentum of a body is equal to the net external force acting on it.

Another useful quantity that we come across is “impulse”. “Impulse” is the product of the net external force acting on a body and the time for which the force is acted.

If a force “F” acts on a body for “t” seconds, then Impulse I = Ft.

In fact, this is also equal to the change in the momentum of the body. It means that due to the application of force, if the momentum of a body changes from “P” to “P ' ”, then impulse,
I =  P ' - P.

For the same change in momentum, a small force can be made to act for a long period of time, or a large force can be made to act for a short period of time. A fielder in a cricket match uses the first method while catching the ball. He pulls his hand down along with the ball to decrease the impact of the ball on his hands.



In a cricket match, when a batsman hits a ball for a six, he applies a large force on the ball for a very short duration. Such large forces acting for a short time and producing a definite change in momentum are called “impulsive forces”.

Derivation of Newton’s Second Law of Motion
Newton’s second law of motion states that the rate of change of momentum of an object is Proportional to the applied unbalanced force in the direction of force.

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
The change in momentum = mv – mu
The change in momentum = m × (v – u).
The rate of change of momentum = m × (v - u) t
 
Or, the applied force,

 F ∝ m × (v - u) t    
 F = km × (v - u) t   = kma     ---------------------------- (i)
 
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 m s-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. (i)

F = ma 
The unit of force is kg m s-2 or newton, represented as N.

Derivation of Newton’s first law of motion from Newton’s Second Law of Motion
Newton's first law states that a body stays at rest if it is at rest and moves with a constant velocity if already moving, until a net force is applied to it. In other words, the state of motion of a body changes only on application of a net non-zero force.

Newton's second law states that the net force applied on a body is equal to the rate of change in  its momentum. Mathematically,
 
      F d p dt    
 
Where, F is the net force, and p is the momentum. Now, we can write the same as:
 F =   d p dt d( mv ) dt
 ⇒  F  = d(m v ) dt    
⇒   F → = md v dt    
                
So, if the net force, F is zero, change in the value of v is be zero i.e., a body at rest will be at rest and a body moving with constant velocity will continue with the same velocity, until a net force is applied. This conclusion is similar to the Newton's first law of motion.
 
Thus, we can derive Newton’s first law of motion using Newton’s second law of motion.
 

Derivation of Newton’s third law of motion from Newton’s second law of motion
Consider an isolated system of two bodies A & B mutually interacting with each other, provided there is no external force acting on the system. 

Let FAB, be the force exerted on body B by body A and FBA be the force exerted by body B on A.
 
Suppose that due to these forces FAB and FBA, dp1/dt and dp2/dt be the rate of the change of momentum of these bodies respectively. 

Then, FBA dp1 dt     ---------- (i) 
     => FAB =   dp2 dt     ---------- (ii) 

Adding equations (i) and (ii), we get, 
FBA + FAB =   dp1 dt +  dp2 dt  

⇒ FBA + FAB = d(p1 + dp2) dt     

If  no external force acts on the system, then 

d(p1 dp2)dt = 0

⇒ FBA + FAB = 0
⇒ FBA = - FAB---------- (iii) 

the above equation (iii) represents the Newton's third law of motion (i.e., for every action there is equal and opposite reaction)...

Videos

Activities


Activity 1

mtchs.org has developed an interactive online simulation which explains how force accelerates the a wooden block. This simulation plots a Velocity - Time graph that is useful to analise the motion of the block. Using this mass of the block and applied force and surface friction can be changed to relate them with the acceleration caused in the block.

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Activity 2

Science-animations.com has developed an interactive online simulation which explains the concepts of momentum, impulse. Using this virtual activity one can better understand the concepts.


Go to Activity

Activity 3

Amrita.olabs.co.in has developed an interactive online simulation which explains how force accelerates a toy cart like a toley. Using this virtual activity the distance - Time graph is plotted and Neton's second law of motion can be verified.

Got to Activity

Activity 4

walter-fendt.de has developed an interactive online Java applet which simulates an air track glider setup, as it is used for experiments on constantly accelerated motion.

To use this applet mass of the wagon, the value of the hanging mass and the coefficient of friction (within certain limits) can be changed.

Go to Activity

References

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