Momentum and its Conservation Formula

In physics, the great scientist Isaac newton had a major impact and in Newtonian mechanics, momentum is a vector quantity that is the product of the mass and velocity of an object or particle. Momentum is measured in the standard unit of a kilogram – meter per second (kg m / s or kg m s ^-1). A quantity that has magnitude but no particular direction is described as scalar whereas a quantity that has magnitude and that acts in a particular direction are described as a vector quantity.

Momentum

“Mass in motion” is a short and accurate statement for the definition of momentum. All objects have mass so if a non-zero mass object is moving, then it has some momentum. The amount of momentum that an object has mostly depends upon two variables,

1. How much stuff is moving?
2. How fast the stuff is moving?

It depends directly upon the mass and velocity. In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object.

Momentum= mass × velocity

In physics, the symbol of the quantity momentum is the lower case p. Thus, the above equation can be rewritten as:

p = m × v

Direction of momentum

Although the direction of the momentum can be expressed in various ways, depending on the number of dimensions involved, it is stated that the “direction of momentum is same as the direction of velocity”. This principle can be better understood by the following example,

Momentum is directly proportional to the velocity, let us take a 1,000-kg truck moving at 20m/s with respect to the surface of a highway , travelling northward. If the truck is driven, then the momentum of the truck is relative to the body of the person driving the truck which is zero. And if person stands by the side of the road, the momentum of the truck relative to that person is 20,000 kgm/s northward.

Types of momentum

Depending upon the nature of motion momentum can be described in following major types,

• Angular Momentum

It is obtained by multiplying a body’s mass by its angular velocity. This means that a single body can have two types of angular momentum . For example, planetary bodies such as Earth have a first momentum that is calculated from the result of its  motion in relation to the Sun and then an additional momentum calculated from the velocity of its spin on its own axis.

• Linear Momentum

It is also known as force, which is the quantity of mass associated with a body that moves along a straight path . An outside object , with its own force can change the trajectory of an object with a linear momentum. For example, if you are running forward and dog runs into you by accident, your trajectory will be changed, and you may fall however you should not be hurt too badly because the momentum of the dog was similar to yours. however, if you hit by truck which has higher linear momentum because of its high weight , you will be lucky to survive. That is because the truck’s force is higher than yours.

• Conceptual Momentum

The everyday meaning of momentum is relatively consistent with its specific meaning, regardless of the type of momentum.

Conservation of momentum and its formula 

Momentum is determined by the product of the mass of the object and its velocity. For two bodies in motion, if both the objects are not experiencing any external forces, then the total momentum of the objects before and after interaction/collision will be the same , this relation under the fundamental law of physics is known as “the conservation of momentum”.

This fundamental law of physics can be applied to explain the phenomenon of collision and explosion. Conservation of momentum can be described by,

P₁ (before) + P₂ (before) = P₁ (after) + P₂ (after)

This equation is valid for the object that undergoes collision. Illustration,

Derivation of conservation of momentum

Let’s consider a situation wherein a truck of mass m₁ , velocity u₁ is moving towards a car of mass m₂ and velocity u₂. Then total momentum of car and truck together is,

momentum = m₁u₁ + m₂u₂.

Now suppose the car and truck collide for a short time t,  and as a result velocities of both will change. So now the velocity of the truck and car become v₁ and v₂ respectively. However , their mass remains the same. Hence now the total momentum = m₁v₁ + m₂v₂.

Total momentum before collision = m₁u₁ + m₂u₂

Total momentum after collision = m₁v₁ + m₂v₂

Acceleration of car (a) = (v₂ – u₂)/t

Also, as F = ma

F₁ = Force exerted by truck on the car.

F₁ = m₂(v₂ – u₂)/t

Acceleration of truck = (v₁ – u₁)/t

F₂ = m₁(v₁ -u₁)/t and F₁ = -F₂

m₂(v₂ – u₂)/t = -m₁(v₁ – u₁)/t

m₂v₂ – m₂u₂ = m₁v₁ + m₁u₁

Or

m₁u₁ + m₂u₂ = m₂v₂ + m₁v₁

Sample problems

Question 1: A shell is fired from a gun with a velocity of 300m/s making an angle 60^o with the horizontal. It explodes into two fragments when it reaches the highest position. The ratio of the masses of the two pieces is 1:3. If the smaller stops immediately after the collision. find the velocity of the other.

Solution: 

Velocity at the highest point = 300 × cos 60° = 150m/s

Using momentum conservation, 150 × m = 3m/4 × v

v= 200m/s

Question 2: There are cars with masses 2 kg and 5 kg respectively that are at rest. A car having the mass 5 kg moves towards the east with a velocity of 5 m.s^-1. Find the velocity of the car with mass 2 kg with respect to ground.

Solution: 

m₁ = 2 kg

m₂ = 5 kg

v₁ = ?

v₂ = 5 m.s^-1

We know from the law of conservation of momentum that,

P_initial= 0, as the cars are at rest

P_final = p₁ + p₂

P_final = m₁v₁ + m₂v₂

= 2 kg × v₁ + 5 kg × 5 m.s^-1

P_i = P_f

0 = 2 kg . v₁ + 25 kg.m.s^-1

v₁ = 12.5 m.s^-1

Question 3: Suppose A 800-kg car moving with a velocity of 10 m/s hits a 2000-kg parked truck. The impact causes the 2000-kg car to be set in motion at 2 m/s. Assuming that momentum is conserved during the collision, determine the velocity of the car immediately after the collision.

Solution:

Momentum of car before collision = 800 × 10 kg.m/s

Momentum of truck before collision = 0 kg . m/s

Total momentum before collision = 8000 kg.m/s

Momentum of car after collision = 800 × v₁ kg.m/s

Momentum of truck before collision = 2000 × 2 kg . m/s

By applying the conservation of momentum :

8000 kg.m/s = 800kg . v₁ + 4000 kg.m/s

v₁ = 4000kg.m.s / 800kg

v₁ = 5m/s 

Hence, speed of car after collision will be 5m/s.

Question 4: List the formula of conservation of momentum along with some examples.

Answer:

The formula of conservation of momentum:

The total momentum of bodies before collision = total momentum of bodies after collision.

Phenomenon which obey conservation of momentum:

• Gun and bullet system
• Collision of vehicles
• Air balloon system etc

Question 5:  Explain the working of Gun-Bullet system with the concept of conservation of momentum.

Answer: 

As per the concept of conservation of momentum, the momentum lost by the primary object is strictly adequate to the momentum gained by the secondary object.  In this case, if a gun exerts a force on a bullet when firing it forward then the bullet will exert an equal force within the other way on the gun causing it to maneuver backwards or recoil. Although the action and reaction forces are equal in size the effect on the gun and therefore the bullet aren’t an equivalent since the mass of the gun is far greater than the mass of the bullet. The acceleration of the bullet while moving along the barrel would be much greater than the acceleration of the gun (acceleration = force mass).

Question 6: Law of conservation of momentum is related to Newton’s laws of motion. Name that Newton’s law of motion?

Answer: 

The Law of conservation of momentum is related to Newton’s third law of motion i.e. every action has equal and opposite reaction. For example, when a person punches on the wall, the wounds are inevitable, this is due to the equal impact given to the person’s hands.

Question 7: If a ball is projected upward by a player from the ground with ten units of momentum, what is the amount of momentum of recoil of the Earth? Why player don’t feel that recoil?

Solution: 

The earth recoils with amount of 10 units of momentum. Since the mass of the Earth is extremely very large, the recoil velocity of the Earth is too small to feel. Therefore, players or people, in general, do not feel that recoil.