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Law of Conservation of Linear Momentum is one of the fundamental laws of Physics. It applies to all objects in linear motion (moving in a straight line). To understand the Law of Conservation of Linear Momentum, one must be familiar with the concept of Linear motion and Linear momentum and its principles.
- When and where this law works, how it affects the motion of a body, exceptions, mathematical derivation of the law, some common day-to-day examples, and practical applications of the Law of Conservation of Linear Momentum.
- Linear Momentum Formula is the product of the mass of the moving object and its velocity.
- So, according to this law, the linear momentum of any moving object is always conserved in the absence of any external force.
- Thus, a body moving in a straight line continues to move in the same direction until an external force is applied to it.
Read Also: Conservation Laws in Physics
Key Terms: Linear Momentum, Mass, Velocity, Newton’s Third law of Motion, Centre of Mass, One-Dimensional Collision
What is Linear Momentum?
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Linear Momentum is the product of the mass of the object and the velocity with which it is moving. It is zero for bodies at rest.
- A stationary body does not have any linear momentum.
- This entity is used to give an idea of how much force is required to change the speed or direction of a moving object.
- The greater the Linear Momentum, the greater the force required to change its direction or speed.
Linear Momentum Formula
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The Linear Momentum is the sum of all the Linear Momentum of all particles constituting the body. In mathematical form, it can be derived as follows:-
| P = m x v |
Where
- P is the linear momentum,
- m is the mass of the object,
- v is the velocity of the object.
This formula is applicable to all objects, regardless of their shape, size, or state of motion. The SI unit of linear momentum is kg m/s. The Linear momentum of a particle is a vector quantity and its unit is kgm/s or kg ms –1.
Law of Conservation of Linear Momentum
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The Second and Third Laws of Motion lead to one of the most important and fundamental principles of physics called the Law of Conservation of Linear Momentum. It can be stated as follows:
| “When no external force acts on a system of several interacting particles, the total Linear Momentum of the system is conserved.” |
The law of conservation of linear momentum is applicable only under the following conditions:
- The body should be in Motion.
- The direction of motion should be constant.
- The linear momentum of the system is conserved and not of the individual particles.
- The momentum of individual particles may change with time, but the sum of the momentum of all the particles remains constant.
- The Law of conservation of linear momentum is equally applicable to all kinds of collisions occurring in nature.
- The law of conservation of Linear Momentum is a concrete and fundamental law that is not affected by Temperature changes, pressure conditions, etc.
Read More: Derivation of Equation of Motion in Physics
Law of Conservation of Linear Momentum Formula
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Law of Conservation of Linear Momentum is universal and applies to bodies moving in a straight line.
- In the case of colliding bodies, this law applies to both Elastic Collision and Inelastic Collision.
- In every phenomenon or change, the initial momentum of a system and the final momentum remain the same if no external Force is applied to the system.
| Law of Conservation of Linear Momentum: Initial Momentum (Pi) = Final Momentum (Pf) |
Derivation of the Law of Conservation of the Linear Momentum
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The Derivation of the Law of Conservation of Linear Momentum can be described mathematically as follows:
Consider two bodies A and B of masses m1 and m2 moving in the same direction along a straight line with velocities u1 and u2 respectively (u1 >u2). They collide for a time interval ∆ of t. After collision let their velocities be v1 and v2.
During a collision, body A exerts a force FBA On body B. From Newton's Third Law of Motion, the body B also exerts a force FAB on the body A in the opposite direction,
so that:
FAB = ̶ FBA
Impulse of FAB = FAB ∆t = change in momentum of A
= m1v1 ̶ m1u1
Impulse of FBA = FBA ∆t = change in momentum of B
= m2v 2 ̶ m2u2
But FAB ∆t = ̶ FBA ∆t
= m1v1 ̶ m1u1 = ̶ ( m2v2 ̶ m2u2)
= m1v1 + m2v2 = m1u 1+ m 2u2
Total Momenta before Collision = Total Momentum after Collision
One-Dimensional Motion and One Dimensional Collision
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One-dimensional motion refers to the motion of an object along a straight line.
- In this type of motion, the object's position is described by a single coordinate, usually measured from a fixed reference point.
- The motion can be either uniform, where the object moves with a constant velocity, or non-uniform, where the velocity changes with time.
One-Dimensional Collision is a collision in which two objects collide and move along a straight line before and after the collision.
- In this type of collision, the motion of the objects can be analyzed in one dimension, along the line of collision.
- In a one-dimensional collision, the total momentum of the system is conserved, which means that the sum of the momenta of the two objects before the collision is equal to the sum of their momenta after the collision.
- However, the kinetic energy of the system may not be conserved, as some of it may be converted into other forms of energy, such as heat or sound.
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For two bodies, A and B, one-dimensional collisions can happen in any of the following ways:
- Body A is moving in a straight line and body B is stationary. After the collision, both bodies start moving in the same direction.
- Both bodies are moving in a straight line but in opposite directions. After the collision, they start moving in the same direction.
- Both bodies are moving in a straight line and in the same direction. After the collision, they start moving in the same direction.
In all three cases mentioned, the total momentum of both bodies before collision (initial momentum) is always equal to the total momentum after collision in the absence of any external force.
This can be expressed mathematically as:-
Initial Momentum= Final Momentum
Pi = Pf
For Body A and Body B, the law of conservation of linear momentum can be written as follows:-
- Total Momentum before Collision= MAUA + MBUB
- Total Momentum after Collision= MAVA + MBVB
Therefore, In
MAUA + MBUB = MAVA + MBVB
where
- MA = Mass of body A
- UA = Initial velocity of body B
- MB = Mass of body B
- UB = Initial velocity of body B
- VA = Final velocity of body A
- VB= Final velocity of body B
Also read: Angular Momentum
Collision in Two Dimensions
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Suppose a body of mass m 1 moves along the X- axis with velocity u 1 collides with another particle of mass m 2 at rest. After collision let the two particles move with velocities v 1 and v 2, making angles θ1 and θ2 with X- the axis.
After the collision, the rectangular components of the momentum of M1 are
- m1 v1cos θ1 along the X- axis.
- m1v1sin θ1 along the Y- axis.
After the collision, the rectangular components of the momentum of M2 are
- m2u2cos θ2 along the X- axis.
- m2u2sin θ2 along the Y- axis.
Applying the principle of conservation of momentum along the X- axis, we get
m1u1 = m1v1 cos θ1 + m2v2 cosθ2
The initial momentum of m1 and m2 along the Y- axis is zero. Applying the principle of conservation of momentum along the Y- axis, we get
0= m1v1 sinθ1 ̶ m2v2 sinθ2
Law of Conservation of Linear Momentum Examples
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The examples of the law of conservation of linear momentum are as follows:
- When a man jumps out of a boat to the shore, the boat moves slightly away from the shore. Initially, the total momentum of the boat and the man is zero. As the man jumps from the boat to the shore, he gains momentum in the forward direction. To conserve momentum, the boat also gains equal momentum in the opposite direction. So the boat moves slightly.
- When a person hits a ball on the wall, it bounces back. The reason behind this is the Law of conservation of Momentum. The ball has some momentum and thus hits the wall with some force. According to Newton third law of motion
- The wall also exerts some force back onto the ball and the momentum of this force is equal to the initial momentum of the ball. Hence, the ball bounces back to the thrower.
- A cannon fires a cannonball. The cannonball moves in one direction with a certain momentum, and the cannon recoils in the opposite direction with equal and opposite momentum. The total momentum of the system (cannon + cannonball) is conserved.
- A rocket launches into space. As it burns fuel, it expels exhaust gases out of the back of the rocket, which creates forward momentum for the rocket. The total momentum of the rocket and its exhaust gases is conserved.
- A car collides with a wall. Before the collision, the car has a certain momentum in a certain direction. After the collision, the car comes to a stop, and its momentum is transferred to the wall in the form of a dent. The total momentum of the system (car + wall) is conserved.
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Applications of the Law of Conservation of Linear Momentum
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Law of Conservation of Linear Momentum have its practical applications as follows:
- Automotive Safety: The law of conservation of linear momentum is essential in designing and testing automotive safety features such as seatbelts, airbags, and crumple zones and are designed to reduce the force of impact during a collision.
- Rocket Propulsion: The law of conservation of linear momentum is crucial in the design and operation of rockets. It works by expelling mass in one direction, causing the rocket to move in the opposite direction.
- The total momentum of the rocket and the expelled mass remain constant, allowing the rocket to accelerate without violating the law of conservation of momentum.
- Sports Equipment: The law of conservation of linear momentum is important in designing sports equipment such as balls, bats, and golf clubs and is designed to transfer momentum efficiently from the athlete to the object and give optimal performance.
- Particle Physics: The law of conservation of linear momentum is a fundamental principle in particle physics and is used to describe the behavior of subatomic particles and the interactions between them.
- Astronomy: The law of conservation of linear momentum is essential in understanding the motion of celestial bodies.
- Industrial Processes: The law of conservation of linear momentum is used in various industrial processes, such as the movement of fluids through pipes and the transport of materials on conveyor belts.
Things to Remember
- Momentum is a vector quantity that is defined as the product of an object's mass and velocity.
- The Momentum of an object is directly proportional to its velocity and its mass.
- The Law of Conservation of Linear Momentum states that the total Momentum of a system remains conserved if no external force is applied to it and applies only to bodies moving in a straight line (linear motion).
- The law of Conservation of linear Momentum applies to both elastic and inelastic collisions.
- Linear Momentum is the product of the Mass and Velocity of the object.
- In any collision, Total Momentum before the collision is always equal to Total Momentum after Collision. Pi = Pf
-
For Collisions occurring in one dimension, the principle of Conservation of Linear Momentum m1v1 + m2v2 = m1u1 + m2u2
Sample Questions
Ques. A Meteorite burns in the atmosphere before it reaches the earth’s surface. What happens to its Momentum? (2 Marks)
Ans. When a meteorite enters the earth’s atmosphere, its motion is due to the force of gravity of the earth.
Throughout its fall its momentum changes but the momentum before burning and after burning remains conserved. So, the falling of a meteorite is also affected by the Law of Conservation of Linear Momentum.
Ques. Why is it advisable to hold a gun tightly to one’s shoulder when it is being fired? (2 Marks)
Ans. The reason is that whenever a gun is shot, according to the; Law of Conservation of Linear Momentum, the gun is pushed back with a momentum equal to the momentum of the bullet. So, the recoiling gun can hurt the shoulder. If the gun is held tightly against the shoulder, the body and the gun will constitute one system. The total mass becomes large and the recoil velocity becomes small.
Ques. Why does a heavy rifle not kick in as strongly as a small rifle using the same cartridge? (3 Marks)
Ans. A heavy rifle does not kick in as strongly as a small rifle because, while shooting, both the rifles follow the Law of conservation of linear momentum as both the rifles have the same cartridge
Accordingly, Initial momentum = Final momentum
For a rifle: MV= mv
- M = mass of the rifle
- V = recoil velocity of the rifle
- m = mass of the bullet
- v = velocity of the bullet
Recoil velocity of a Rifle, V= – \(\frac{m}{M}\)v i.e. V ∝ \(\frac{1}{M}\)
Thus, the recoil velocity of the heavy rifle is smaller than the light rifle, so it does not kick strongly.
Ques. A car of mass 1000 kg traveling at 32 m/s dashes into the rear of a truck of mass 8000kg moving in the same direction with a velocity of 4 m/s. After the collision, the car bounces off with a velocity of 8 m/s. What is the velocity of the truck? (5 Marks)
Ans. Solution for the car:
For the car: m1 = 1000 kg; u1 = 32 m/s; v1 = - 8 m/s
For the truck: m2 = 8000 kg; u2 = 4 m/s; v2 = ?
By conservation of linear momentum: M1v1 + m2v2 = m1u1 + m2u2
Total momentum before collision= Total momentum after the collision
1000×-8 + 8000×v2=1000×32 + 8000×4
= 64000 + 8000 = 8000v
= v2 = \(\frac{72000}{8000}\) = 9 m/s
Hence, the final velocity of the truck is 9 m/s.
Ques. A body of mass 1kg initially at rest explodes and breaks into three fragments of masses in the ratio of 1:1:3. The two pieces of equal masses fly off perpendicular to each other with a speed of 30 m/s each. What is the velocity of the heavier fragment? (5 Marks)
Ans. The mass of the body:
Here, m1 + m2 + m3= 1kg
As m1 : m2 : m3 = 1:1:3
m1 = m2 = 0.2 kg; m3 = 0.6 kg
v1 = v2 = 30 m/s; v3 =?
Applying the law of conservation of Momentum to Momenta along the horizontal direction;
m3v3= m1v1 cos 45° + m2v2 cos 45°
0.6v3 =0.2×30×0.707 + 0.2 ×30 ×0.707
v3 =2×0.2 30×0.707 / 0.6=14.14
Hence, the final velocity of the heavier fragment is 14.14 m/s.
Ques. A 30g bullet leaves a Rifle with a velocity of 300 m/s and the Rifle recoils with a velocity of 60m/s. Find the Mass of the Rifle. (3 Marks)
Ans. M = ?; V= 60 m/s
m = 30 g; v= 300 m/s
MV = – mv
M= – \(\frac{m}{V}\)v
M= \(\frac{0.03}{60}\) 300 =15
Hence, the mass of the rifle is 15kg.
Ques. It is easy to catch a Table Tennis ball than a cricket ball even when both are moving with the same velocity. Why? (1 Mark)
Ans. Due to its small mass, the momentum of the table tennis ball is much smaller than that of the cricket ball of the same velocity. Less force is required to stop the tennis ball than the cricket ball. Hence, it is easy to catch a table tennis ball than a cricket ball.
Ques. A light body and a heavy body, both have the same kinetic energy. Which one will have the greater momentum? (2 Marks)
Ans. Kinetic energy:
K= \(\frac{1}{2}\) mv =\(\frac{(mv)^2}{2m}\)= \(\frac{p^2}{2m}\)
p= √ 2mK i.e; p ∝ √m
Thus, the heavier body has greater momentum than the lighter body.
Ques. Is the explosion of a bomb follow the Law of Conservation of Linear Momentum? Explain. (3 Marks)
Ans. Before the explosion, suppose the bomb is at rest. Its total momentum is zero. As it explodes, it breaks up into many parts of masses m1, m2, m3, etc., which fly off in different directions with velocities v1, v2,v3, etc.
The different parts have definite momenta – m1v1, m2v2, m3v3, etc.
The momenta of the various parts can be represented by the sides of a closed polygon taken in the same order. This indicates that the total momentum after the explosion is zero ie., momentum is conserved. If a bomb explodes into two parts, it will fly off in opposite directions with equal momentum.
Ques. Is the Law of Conservation of Linear Momentum affected by any change in the physical conditions of the system? (5 Marks)
Ans. The law of conservation of linear momentum states that the total momentum of an isolated system remains constant if no external forces act on it.
However, if the physical conditions of the system change, such as if an external force is applied to it, the law of conservation of linear momentum may no longer hold true.
- In such cases, the change in momentum of the system will be equal to the net external force acting on it multiplied by the time interval during which the force is applied.
- Additionally, the law of conservation of linear momentum applies only to systems that are isolated, meaning that no external forces act on them.
- If external forces are present, such as friction or air resistance, they can cause the total momentum of the system to change.
- However, in many cases, the effects of these external forces can be taken into account and the law of conservation of linear momentum can still be applied to the system.
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