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Saturday, 31 March 2018

conservation of momentum formula

conservation of momentum formula

Today our topic is conservation of momentum very important topic. We will discuss this topic from basic concept to depth concept of conservation of momentum.First we will discuss what is momentum and then how momentum of a system is conserved by taking practical example.
In previous topic we have already learn about centre of mass of a system, So that concept of centre of mass also be use in the conservation of momentum.
So i will suggest you to go through topic of centre of mass for easy understand, You can refer previous post Centre of mass important point  to understand concept of centre of mass. Now lets start conservation of momentum.  




What is momentum ?  

                                                                     
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                                                     Conservation of momentum        
                                                                     
As from the above figure momentum. Bicycle and truck are moving towards you as shown  then which one you will try to stop.
I am sure you will try to stop bicycle not truck.But why the reasoning behind this is momentum. The momentum of truck is more than bicycle.
Momentum is simply mean mass in motion. Momentum of truck is more because it is very massive. Now momentum of bicycle is less because it has less mass. But it does not mean that momentum only depend upon mass. It depend upon velocity also.

Hence momentum mathematical relationship with mass and velocity is define as  momentum = mass*velocity or denoted p = mv.
Where p is momentum m is mass in kg and v is velocity. 
You should remember that velocity is define as speed with direction, If an object has large speed it also has large velocity. we know velocity is vector quantity so momentum is a vector quantity. 

From the above momentum equation  p = mv we see that if you vary the mass and velocity momentum will change.
If you increase either of variable mass or velocity momentum will go up. So truck momentum is more than bicycle. This also means that an object at rest does not have momentum. Velocity of an object at rest is zero so no movement.

So in order to an object having momentum it must be moving. I hope you have understand momentum.
Now let me discuss more interesting about momentum, An important point to note about momentum. It is a vector quantity this means that it has magnitude as well as direction.




The direction of momentum is same as the direction of velocity vector. But in order to fully describe the momentum of an object must include direction.

Now look at some  example of momentum for better understanding. An object of any size can have large momentum whether its size is large or small but can have large momentum.
Because its depends upon the speed of an object as well its mass.
If its speed is more then its momentum is more and if its mass is large its momentum is large. Now you think that all object which have greater mass have greater momentum but it is not true for all cases .

Let us discuss in this case a super tanker ship is coming to the port not be travelling very fast through the water but it is massive so it will be easy to stop at port safely.

Lets understand momentum with daily life example . A truck of full load has a large mass and must slow down long before a stop light because even with a small velocity, It has a large momentum and is difficult to stop.
A bullet, although small in mass, has a large momentum because of an extremely large velocity.

What is Conservation of momentum ? 

As we have seen mathematical relation of momentum p = mv now differentiate this equation on both side with respect to time then will get.
dp/dt  = mdv/dt  here mass is taken as constant, Now we know that dv/dt is acceleration but remember which acceleration this is centre of mass acceleration if you are not digesting refer previous centre of mass post.
Hence we can write dp/dt  = ma and from Newton's second law ma is Fnet or we can write 

dp/dt  = Fnet  now try to understand physical meaning of this equation.
Rate of change of momentum is net force on the system. Its physical meaning is if force is applied on a body then its momentum will be greater in the force applied direction.
                                                               
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See the Above picture for conservation of momentum suppose we have a system as shown in which there are some objects which are moving as shown in figure.
As we have learn the system will have a centre of mass as shown by pink mass and it will also have a velocity which is known as velocity of centre of mass Vcm.

Now we are saying that the external force on the system is zero Fnet =0, Internally the object or particle can have apply force. But externally there is no any force acting on the system.
So as external force is zero its mean that acceleration of centre of mass is also zero acom =0, Then Vcm will be constant Vcm = constant, magnitude as well as direction constant.
Hence its mean that momentum of the system is constant.

So it is very important conclusion derived, If external force on the system is zero that is Fnet =0 then momentum of the system is conserved or constant means initial and final momentum of the system will be same.

Conservation of momentum definition 

Principle of conservation of momentum states that if external force on a system is zero,Momentum of system remains constant or conserved.

Now we will see some numerical questions so that your concept and doubt may clear, how to apply conservation of momentum formula. see below questions.
                                                                     
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From the question considering two men together as a system and it is given initially the system is at rest. So it is clear that external force on the system is zero hence Fnet = 0.

Hence total momentum of the system should remain constant that p = constant ,so we can write Pf = Pi 
Now initial momentum of the system is zero both men are in rest initially 
so  Pi  = 0+0 = 0
Now final momentum of the system is Pf =  100*v - 50*10  ( negative for 50kg because momentum is vector quantity consider with sign it is in negative direction)




Now equate Pf = Pi, Hence  100v-500 =0 or  v = 500/100 = 5m/s  v= 5m/s in right direction.

So you have seen that if external force is zero then momentum is conserved, and how to apply to solve the numerical question.
                                                                        
conservation of momentum formula,what is momentum,momentum formula,conservation of momentum example

From the above picture question Q1.
We have to find velocity of car when man is walking above the car with 2m/s suppose friction between ground and car is negligible .

Here external force is zero, Fnet = 0 so momentum will remain conserved.
Final momentum = Initial momentum 
initial momentum is zero Pi = 0 because initially man is standing and car is at rest 
now for final momentum let v be the velocity of car, important point is consider velocity with respect to ground.

Pf = m*2  -10m(2-v) = 2m - 20m-10mv  = -18m -10mv  Hence 

Pf = Pi
 -18m -10mv  = 0
v  = -18m/10m  = -1.8m/s.

Now you can try for second question Q2 .
Hints before one mass is moving and other mass is at rest and after collision both stick together then find final mass velocity.

Now we will continue in next post . I hope that you have enjoyed learning conservation of momentum formula. If you like comment and share thanks for sharing and reading learn and grow.

Dated 9th Nov 2018   





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