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Conservation of linear momentum law states that: When two (or more) bodies act upon one another, their total momentum remains constant (or conserved) provided no external forces are acting.
- The decrease in the momentum of one body is equal to the increase in the momentum of another body.
- It can also be stated as “Momentum can neither be created nor destroyed”.
- The law of conservation of linear momentum is also called the principle of conservation of linear momentum.
- The law of conservation of linear momentum is in accordance with Newton’s Third Law of Motion, “Every action has an equal and opposite reaction”.
Read more: Kepler’s law
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KeyTerms: Collision, Recoil of gun, Impulse force, Linear momentum, Mass, Velocity
Linear Momentum
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An object's momentum is determined by multiplying its mass by its velocity. It is represented by "p" and is a vector quantity. If "p" represents momentum, an object's mass is "m," and its velocity is "v."
Then,
| p = m x v |
- The momentum is directly proportional to the mass and velocity of an object.
For example: A lightweight vehicle and a heavy-weight vehicle are parked on a horizontal road. Much greater force is needed to push the heavy vehicle than the lighter one to bring them to the same speed at the same time.
- A body's mass plays a crucial role in determining how force affects its motion.
For example: If two stones, one light and the other heavy, are dropped from the top of a building, a person on the ground will find it easier to catch the light stone than the heavy stone.
- Speed is another important parameter to consider.
For example: A bullet fired by a gun can easily pierce human tissue before it stops, resulting in a casualty. When fired at a reasonable pace, the same bullet will cause no harm. Thus for a given mass, the greater the speed, the greater the opposing force needed to stop the body in a certain time.
Momentum, which is the sum of mass and velocity, is definitely a significant motion-related variable.
- The greater the change in the momentum in a given time, the greater the force that needs to be applied.
Read more:
| Relevant Concepts | ||
|---|---|---|
| Law of Inertia | Newton Second Law of motion | Elastic Collision |
| Newton First Law of Motion | Inelastic Collision | Relative speed |
Derivation of Conservation of Linear Momentum
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Consider the collision of 2 objects:
- Mass: m1 and m2.
- Initial velocities before collision: u1 and u2.
- Final velocities after the collision: v1 and v2.
- Force applied by the second object on the first: F21.
- Force applied by the first object on the second: F12.
- Time of contact: t.
According to Newton’s Third Law of Motion: “Every action has an equal and opposite reaction”.
So,
F21= – F12. (1)
By definition, the impulse of force F21 is equal to change in momentum of the first object
F21. t = m1v1 − m1u1 (2)
The second object's changed momentum is equal to the impulse of force F12.
F12. t = m2v2 − m2u2 (3)
From (1) F21 = − F12
F21. t = − F12. t
m1v1 − m1u1 = − (m2v2 − m2u2)
m1u1 + m2u2 = m1v1 + m2v2
This suggests that momentum is conserved during the collision.
Application of Conservation of Linear Momentum
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Linear momentum plays a significant role in rocket propulsion and in recoil of guns. They include:
Rocket Propulsion
The rocket contains chemicals that burn and eject gasses at high velocity. These gasses pass out through the tail nozzle in the downward direction and subsequently, the rocket moves upward to balance the momentum of gasses. Though the mass of gas emitted is comparatively small, their velocity adds up to the momentum. AN equal momentum is imparted to the rocket in the opposite direction and so despite its great mass, the rocket moves upward with a high velocity.
Rocket propulsion
Recoil of Gun
- Before firing, the pistol and bullet are at rest. So, the initial momentum of the bullet and gun is zero, as their initial velocities are zero.
- When a bullet is fired from a gun, the momentum of the bullet is given by the formula:
Momentum of bullet = Mass of bullet x Velocity of the bullet.
- The bullet imparts an equal and opposite momentum to the gun as a result of which the gun jerks backward (recoil). The backward velocity of the gun is called recoil velocity.
Momentum acquired by Gun = Mass of Gun x Recoil velocity of Gun.
- According to the law of conservation of linear momentum:
The momentum of the bullet = Momentum of the Gun.
| Mass of bullet x Velocity of bullet = Mass of Gun x Recoil velocity of Gun. |
Recoiling gun
Read more: Angular momentum
Things to Remember
- Momentum is the product of mass and velocity. It is a vector quantity.
- Momentum can neither be created nor destroyed.
m1u1 + m2u2 = m1v1 + m2v2
- The force that must be supplied increases as the change in momentum at a particular time increases.
- The law of conservation of momentum can be seen in real life in the recoil of gun, rocket propulsion, inflated balloons, Billiards and Snooker, and Bowling.
Read more: Molecular collision
Sample Questions
Ques. What is Impulse? (3 marks)
Ans. When a large force acts for a very short duration producing a finite change in the momentum of the body. Often, in these situations, the force and the time duration are difficult to ascertain separately.
However, the product of force and time, which is the change in momentum of the body, remains a measurable quantity. This product is called impulse.
| Impulse = Force × time duration = Change in momentum. |
A large force acting for a short time to produce a finite change in momentum is called an impulsive force.
Ques. What is momentum? (2 marks)
Ans. The momentum of an object is defined as the product of its mass and velocity. It is a vector quantity and denoted by “p”.
- If “p” is momentum, m is the mass of an object, and v is the velocity.
Then,
| p = m x v. |
Ques. Find the velocity of a bullet of mass 4 grams that is fired from a pistol of mass 2.5 kg. The recoil velocity of the pistol is 1.5 m.s-1. (5 marks)
Ans. Given,
Mass of the bullet, m1 = 4 gram = 0.004 kg
Mass of a pistol, m2 = 2.5 kg
The velocity of a bullet, v1 =?
Recoil velocity of pistol, v2 = 1.5 m.s-1
Using law of conservation of momentum,
m1u1 + m2u2 = m1v1 + m2v2
Here, Initial velocity of the bullet, u1 = 0
Initial recoil velocity of a pistol, u2 = 0
∴ (0.004 kg)(0) + (2.5 kg)(0) = (0.004 kg)(v1) + (2.5 kg)(1.5 m.s-1)
0 = (0.004 kg)(v1)+(3.75 kg.m.s-1)
v1=-937.5 m.s-1
Hence, the recoil velocity of the pistol is 937.5 m.s-1
Ques. Car A of mass 1500 kg, traveling at 25 m/s collides with another car B of mass 1000 kg traveling at 15 m/s in the same direction. After the collision, the velocity of car A becomes 20 m/s. Calculate the velocity of car B after the collision. (10 marks)
Ans. The momentum of car A (Before Collision)= Mass of Car A × velocity
1500 × 25.
= 37500 kg.m/s
Momentum of car B (Before Collision)= Mass of Car B × velocity of car B.
= 1000 × 15 = 15000 kg.m/s.
The total momentum of car A and car B (before the collision)
= 37500 + 15000
= 52500 kg.m/s.
After the collision, the velocity of car A of mass 1500 kg becomes 20 m/s.
The momentum of car A (after collision) = 1500 × 20 = 30000 kg.m/s.
After the collision, suppose the velocity of car B of mass 1000 kg becomes v m/s. So, the Momentum of car B (after collision) = 1000 × v = 1000 v kg.m/s.
The total momentum of car A and car B (after collision) = 30000 + 1000 v.
Now, according to the law of conservation of momentum:
Total momentum before collision = Total momentum after collision
That is, 52500 = 30000 + 1000 v
1000 v = 52500 – 30000
1000 v = 22500
v = 22500/ 1000
v = 22.5 m/s
Thus, the velocity of car B after the collision will be 22.5 m/s
Ques. What is a collision? (2 marks)
Ans. When two bodies come in direct contact with each other, they exert force on each other in a short period of time.
The energy and momentum of the colliding bodies changes. It may occur through actual physical contact or without any actual physical contact.
Ques. A bullet of mass 10 g moving with a velocity of 400 m/s gets embedded in a freely suspended wooden block of mass 900 g. What is the velocity acquired by the block? (10 marks)
Ans. Mass of the bullet, m1 = 10 g = 10 kg 1000 = 0.01 kg.
The velocity of the bullet, v1 = 400 m/s.
So, the Momentum of the bullet = m1 × v1.
= 0.01 × 400 kg.m/s.
This bullet of mass 10 g gets embedded into a wooden block of mass 900 g.
So, the mass of the wooden block along with the embedded bullet will become
900 + 10 = 910 g.
Thus, Mass of wooden block + Bullet,
m2 = 900 + 10 = 910 g
= 910 kg/ 1000 = 0.91 kg.
The velocity of wooden block + bullet, v2 =? (To be calculated)
So, Momentum of wooden block + bullet = m2 × v2 = 0.91 × v2 kg.m/s.
According to the law of conservation of momentum
m1 × v1 = m2 × v2 or 0.01 × 400 = 0.91 × v2
v2 = 0.01 × 400/ 0.91
= 4.4 m/s.
Thus, the velocity acquired by the wooden block (having the bullet embedded in it) is 4.4 meters per second.
Ques. What are the types of collisions? (3 marks)
Ans. There are 3 types of collision:
- Perfectly Elastic Collision: The kinetic energy before and after the collision is constant. For example, Collision between atomic particles.
- Perfectly Inelastic Collision: After the collision, the two bodies stick together and move with the same velocity. For example, the firing of a bullet embedded in a wooden block.
- Inelastic Collision: The kinetic energy before and after collision is not the same. For example, a Collision between two vehicles on the road.
Ques. What is angular momentum? (3 marks)
Ans. The angular momentum is defined as the property of any rotating object. It is given by the product of the moment of inertia and angular velocity of the rotating object.
- It is a vector quantity and has both magnitude and direction.
- The common examples of angular momentum are the rotation and revolution of Earth.
- Its SI unit is Kgm2s-1.
Ques. What is acceleration? (3 marks)
Ans. Acceleration is the rate of change of velocity. The change in velocity can be increasing velocity or decreasing velocity or a change in direction of motion.
It is equal to the difference between the initial and final velocities divided by the time.
Some examples of acceleration include the moon orbiting around the earth, the falling of an apple, etc. An object is said to be accelerating if it is changing its velocity time and again.
Ques. What are the 3 Newton’s Laws of Motion? (3 marks)
Ans. Newton’s First Law of Motion: A body in a uniform motion or at rest will continue to be in uniform motion or at rest until and unless a net external force acts on it.
Newton’s Second Law of Motion: The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts.
Newton’s Third Law of Motion: Every action has an equal and opposite reaction.
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