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Conservation of Momentum

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Every where you notice the bodies moving with its own momentum. The ice skater, the gymnasts, the bullet getting fired. These all possess their own momentum. So how can you justify that their momentum is conserved or not? Lets see about it in this section.


The Law of  Conservation of Momentum

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It states that the total momentum before collision is equal to the total momentum after the collision. It is a vector quantity and is always directed in the direction of velocity. The unit is kgm/s or Ns. It depends on mass and velocity of the body.
Law of Conservation
Take two bodies A and B in an isolated system of masses m1 and m2 with the initial velocities u1 and u2 collides each other attains final velocities v1 and v2. Thus

Total momentum before collision = Total momentum after collision
Momentum before collision = m1u1 + m2u2
Momentum after collision = m1v1 + m2v2
So the Law of Conservation tells that m1u1 + m2u2 = m1v1 + m2v2

Conservation of Momentum Formula

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Take two bodies A and B of masses m1 and m2 having initial velocities u1 and u2. If these bodies collide for time t and their velocities change and attain v1 and v2 after collision.

From Newton's second law of motion we have
Rate of change of momentum of A, $\vec{F_A}$ = $\frac{\Delta \vec{P_1}}{t}$
Rate of change of momentum of B, $\vec{F_B}$ = $\frac{\Delta \vec{P_2}}{t}$

The Newtons third law is $\vec{F_B}$ = - $\vec{F_A}$

The force exerted by A on B is equal and opposite to the force exerted by B on A
$\frac{\Delta \vec{P_1}}{t}$ = - $\frac{\Delta \vec{P_2}}{t}$

The Momentum $\Delta{\vec{p_1}}$ = m1v1 - m1u1 and $\Delta{\vec{p_2}}$ = m2v2 - m2u2
So m1v1 - m1u1 = -(m2v2 - m2u2)
(m1u1 + m2u2) = (m1v1 + m2v2)
Thus total momentum before collision = total momentum after collision.
The vector sum of all the momenta for a closed system with no external force acting on it is a constant.
p1 + p2 + p3 + p4 + . . . . .+ pi = K, where K is a constant.
The law of conservation of angular momentum states that in a closed system the angular momentum always remains the same till no external torque is acts on the system. If a body of mass m is rotating with velocity v at a distance r from origin carrying momentum p. The angular momentum is given as
L = mvr
As momentum p = mv
L = rp

Example : During skating if the skater folds hands then he is able to have a greater rotational energy at a greater speed as compared to when he opens his hands wide. This tells about conservation of angular momentum.
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The linear momentum conservation is same as newtons first law. It tells that linear momentum will be conserved if no external force acts on it.
For a particle the linear momentum is a vector quantity is defined as
$\vec{p}$ = m $\vec{v}$
The Newtons second law says that
$\vec{F}$ = m $vec{a}$ = m $\frac{d \vec{v}}{dt}$ = $\frac{d \vec{p}}{dt}$
$\vec{F}$ = $\vec{0}$
That implies that momentum p is a constant tells that momentum is conserved.

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Conservation of Momentum Experiment

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The Newtons cradle is known for demonstrating the law of conservation of momentum shown in fig. Here you could see the n number of balls swinging in is equal to the n number of swinging out. This demonstrates the law of conservation of momentum through collision.

Newtons Cradle

The trolley method can also be used to demonstrate the collisions in the law.

Conservation of Momentum Examples

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Here are some examples of conservation of momentum which illustrates you alot more about the how the law works:

You could see the young snooker aiming the yellow ball

Conservation of Momentum Examples
If the white ball collides the yellow one.The white ball loses the momentum whereas the yellow ball gains equal amount of momentum after collision. It tells that the total momentum before collision is same as that after collision.

If you are at shoot with rifle or shotgun you are often told to hold the gun tightly against your shoulder. Why is it so?
Conservation of Momentum
when the bullet is fired it moves in the forward direction and the gun moves backward. It is due to the gaining of some momentum due to its velocity. To nullify this momentum gun moves takes a backward push such that it has same momentum magnitude to that of momentum of bullet with opposite direction.

Conservation of Momentum Problems

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Here are some sample problems on conservation of momentum you can go through it:

Solved Examples

Question 1: A bomb of mass 8 kg initially at rest explodes into two fragments of masses 3 kg and 5 kg. If 3 kg mass moves with a velocity of 4 ms-1, what will be the velocity of 5 kg mass?
Solution:
 
Given: mass m1 = 3 kg and mass m2 = 5 kg
Since the bomb is initially at rest u = 0, v1 = 4 ms-1 and v2 =?
The law of conservation of momentum is
m1u1 + m2u2 = m1v1 + m2v2
0 = 3 $\times$ 4 + 5 v2
0 = 12 + 5v2
v2 = - $\frac{12}{5}$ = - 2.4 ms-1
5 kg mass moves in the direction opposite to 3 kg mass. Hence v2 is negative.

 

Question 2: An archer fires a 1 kg of arrow at 40 m/s. If both archer and bow weighs 70 kg. What will be the velocity of archer after firing the arrow?
Solution:
 
Given: Mass m1 = 70 kg, mass m2 = 1 kg, Initial velocity of archer u1 = 0, initial velocity of arrow u2 = 0, Final velocity of archer v1 = 40 m/s, Final velocity of arrow = ?
Momentum before collision pi = Momentum after collision pf
m1u1 + m2u2 = m1v1 + m2v2
0 = 70 kg $\times$ 40 m/s + 1 kg $\times$ v2
Final velocity v2 = - 2800 m/s.