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Linear Momentum, Law Of Conservation Of Momentum and its application, Practice Problems, FAQs
How do you think rockets or planes change their direction in space? This is completely based on the principle of conservation of momentum. When you fire a bullet from a gun then the gun recoils. This is because a high speed fired bullet has some momentum and to compensate for this momentum the gun acquires recoil speed. We will study conservation of momentum principle and how it is derived using the third law of motion.
Table of Contents
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Linear momentum
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Law of conservation of momentum
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Conservation of momentum using third law
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Application of conservation of momentum
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Practice problems
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FAQs
Linear momentum
Have you ever wondered how a lorry , moving with a greater speed, upon collision with a wall, would do more damage than compared to a lighter car moving slowly? This is because, the lorry being a heavier body, has a higher linear momentum compared to the car. The linear momentum of a body is defined as the product of its mass and velocity. Mathematically, it can be expressed as
- Linear momentum is a vector quantity.
- Its unit is
and dimension is 
.
Law of conservation of momentum
If all the external forces acting on a system add up to zero, the linear momentum of the system remains constant. This is known as the principle of conservation of linear momentum (PCLM).
Mathematically,
When 
Then 
Therefore, 
In any particular direction, if the external forces acting on the system add up to zero, the linear momentum will be conserved along that direction only.
Example: If the external forces acting on the system add up to zero along the x-axis, the linear momentum of the system remains constant along the x-axis.
Conservation of momentum using third law
Newton’s third law states that for a force applied by an object A on object B, object B exerts back an equal force in magnitude, but opposite in direction. This idea was used by Newton to derive the law of conservation of momentum.
Consider two colliding particles A and B whose masses are 
and 
with initial and final velocities as 
and 
of A and 
and 
of B. The time of contact between two particles is given as 
.
Let’s first calculate the initial and final momentum of A and B,
Initial momentum of A, 
Final momentum of A,
Change in momentum of particle A,
Similarly,
Initial momentum of B, 
Final momentum of B, 
Change in momentum of particle B,
Now, from newton's third law of motion,
Hence, if net force on a system is zero then total momentum of the system is conserved.
Application of conservation of momentum
Principle of conservation of momentum has various applications. This principle is used in the satellites to change the orbit of satellites. Using this principle we can analyse different situations happening around us. Let’s consider one example for better understanding. Why do you think guns get recoil after firing bullets from it?
Recoiling of gun: When a bullet is fired from a gun, it exerts forward force on the bullet and the bullet exerts equal and opposite force on the gun. So the net force on the system is zero. Therefore we can apply the law of conservation of linear momentum here. As initially the system was at rest so initial momentum was zero. After firing the bullet, the bullet has some momentum in the forward direction. As initial momentum was zero then final momentum must be zero, according to the law of conservation of momentum. So make the final momentum zero the gun gets recoil speed in backward direction. Due to the high mass of the gun, it moves a little distance backward and gives a backward jerk to the shoulder of the gunman.
Video Explanation: Linear Momentum (Time Stamp: 18.20 to 19.39)
Practice Problems
- A gun fires
bullets per second. The mass of each bullet is 
and the velocity of the bullet when it leaves the gun is 
the force required to hold the gun while firing is?
Solution:
Number of bullets =
Mass of bullet = 
Velocity of bullet =
Force required to hold the gun = number of bullets 
rate of change of momentum
- A body of mass
. A mass of
Solution: As the net external force acting on the mass is zero, we can use the principle of conservation of linear momentum (PCLM).
Therefore,
.
- A bullet of mass
. Calculate the recoil speed of the gun?
Solution:
Select the gun and the bullet as a single system.
Let 
be the recoil velocity of the gun and 
be the velocity of the bullet. As there is no external force acting on this system, we can use the principle of conservation of linear momentum here.
Initial momentum of the system is as follows:
Final momentum of the system is as follows:
Thus,
.
- Derive the law of conservation of linear momentum using Newton’s second law of motion?
Solution:
According to Newton’s second law of motion,
Therefore,
Hence proved.
FAQs
- What is the principle of conservation of linear momentum?
Solution: If all the external forces acting on a system add up to zero, the linear momentum of the system remains constant. This is known as the principle of conservation of linear momentum.
- What is the unit and dimension of momentum?
Solution: Unit of momentum is and dimension is .
- What is the condition under which linear momentum is conserved?
Solution: If all the external forces acting on a system add up to zero then momentum of the system is conserved.
- What is linear momentum?
Solution: The linear momentum of a body is defined as the product of its mass and velocity.
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