Kinetic energy and momentum - Relation with Derivation and Examples

Kinetic energy and momentum - Relation with Derivation and Examples

Aditya Raj Anand

Sunday 3 January 2021

Physics

Kinetic energy and momentum both the terms have relation between them. As we know that kinetic energy is the energy possess due to the momentum or (motion) of the body. So, there is relation between kinetic energy and momentum. Relation has discussed in later part in detail. relation b/w kinetic energy and momentum cannot be understand without their derivation. So derivation has also given in further. But before we established the relation of kinetic energy and momentum, we should know about kinetic energy and momentum. So lets start with Kinetic energy.

What is kinetic energy?

The energy of a body generated by the motion of the particles called its kinetic energy. In other words kinetic energy is due to the motion of the particles. Faster the movement of the body greater the kinetic energy and lower the motion of the body smaller will be kinetic energy.

In above we discussed kinetic energy on the basis of movement or motion of the particles. But Now, we define kinetic energy on the basis of momentum.

Kinetic energy is the energy required to accelerate the body from rest to motion. In other words Kinetic energy is the energy required to change the momentum of the particles so that it can change its velocity.

• Kinetic energy is represent as K.E.
• It is scalar quantity.
• Kinetic energy has no direction.

A moving hammer drives a nail into the wall because of its kinetic energy. on the same way a moving bullet can penetrate even a steel plate due to its kinetic energy. In fact, every object around us which is moving possessed kinetic energy. In other words every object around us has kinetic energy. For example a runner has kinetic energy; a falling stone has kinetic energy etc. But remember when a body is bringing to rest its kinetic energy will be lost.

hence, the formula of kinetic energy is K.E = 1/2 mv2. where m is mass, v is velocity. Thus, a body of mass 'm' moving with velocity 'v' has the capacity of doing work equal to 1/2 mv2 before stops.

What is Momentum?

Momentum is the product of mass and velocity. In other words momentum is directly proportional to the mass and velocity. that means if the mass and velocity of the body increases then momentum also increases and vice-versa.

• Momentum is represent as P.
• It is vector quantity.
• Momentum has direction.
• Formula of momentum is P = mv

A moving object always possess momentum (no matter how the speed of the object) is equal to the product of mass and velocity. So we can say that if moving body increases its velocity then momentum also increases. For example, A moving cricket ball before hits has less momentum then after hits by the batsman. 

Please note that momentum can never be created nor be destroyed but it can be change.

Unit of momentum

As we know that momentum is the product of mass and velocity. 
P = mv
Hence the unit of momentum is Kg.m/s.

Relation between kinetic energy and Momentum

The relationship between kinetic energy and momentum is, kinetic energy depends on its mass and velocity. In the same way momentum also depends on mass and velocity. therefore, heavy bodies moving with high velocities have more kinetic energy and momentum than slow moving bodies of small mass.

In other words kinetic energy is directly proportional to its mass and velocity of the body. On the same way momentum also directly proportional to its mass and velocity. therefore, if the mass of the body is double, its kinetic energy and momentum also get double and vice-versa.

Please note that doubling the velocity has greater effect on the kinetic energy of the body than doubling its mass. But not in the case of momentum.

The mathematical relation between kinetic energy and momentum is, twice of Kinetic energy is equal to the product of momentum and velocity. that is given by,

K.E = 1/2 mv2
K.E = 1/2 Pv
2K.E = Pv

According to this relation if kinetic energy increases, momentum also increases. on the other hand if kinetic energy decreases momentum also decreases.

We have just seen that the kinetic energy of the body of mass 'm' and moving with a velocity 'v' is given by the formula,
Kinetic Energy = 1/2 mv2

From this formula,

• Kinetic energy is directly proportional to the mass of the body.
• Kinetic energy is directly proportional to the square of the velocity of the body.

We have also just seen above that the momentum of a body of mass 'm' and moving with velocity 'v' is given by the formula,
Momentum = PV

Form this formula,

• Momentum is directly proportional to the mass of the body.
• Momentum is directly proportional to the velocity of the body.

Hence, Kinetic energy and momentum have lots of similarities . Both are directly proportional to their masses and velocities.

Derive the relation between kinetic energy and momentum.

Suppose a body of mass 'm' moving with velocity 'v' in a medium. After some time its velocity decreases so that it comes to rest. Final velocity of the body become zero. Let the total distance travelled during the whole journey is 's'.

In going through distance 's' the body has done some work. This work is calculate as:

Work done = Force ✕ Distance

W = F ✕ s

If the body has initial velocity 'v', final velocity 'V', acceleration 'a' and travels a distance 's', then according to third laws of motion:

V2 = v2 + 2as -------------- (1)

In the above example, we have:

Initial velocity of the body = v (supposed)

Final velocity of the body = V = 0 ( the body stops)

Acceleration of the body = -a (due to retardation)

Distance travelled = s (let)

Now, putting these values in equation (1) we get,

(0)2 = v2 - 2as

v2 = 2as ------------- (2)

From Newton's second laws of motion, we have:

F = ma

a = F/m

Again put the value of 'a' in equation (2) we get,

v2 = (2 ✕ F ✕ s) / m

F ✕ s = 1/2 mv2

As we discuss above kinetic energy is equal to the amount of work done by body that is F ✕ s.

Kinetic energy = F ✕ s = 1/2 mv2

Kinetic energy = 1/2 mv2 -------------- (3)

Now, 

As we know the body of mass 'm' moving with velocity 'v' has a momentum equal to the product of mass and velocity.

Therefore, Momentum = Mass ✕ velocity

P = Mass ✕ velocty

P = mv

Kinetic energy = 1/2 mv2 [ from equ. (3) ]

K.E = 1/2 mv2 ✕ m/m  [ multiply and divide by mass ]

K.E = 1/2 m2v2 / m

K.E =  p2 / 2m  [ P = mv ] 

Hence, K.E = p2 / 2m or 2 K.E = p2 /m

Therefore by this derivation we can also say that there is also a directly relation between kinetic energy and momentum. Kinetic energy is directly proportional to the square of the momentum and inversely proportional to the mass of the body.

Examples of relation between kinetic energy and momentum

Consider a body of mass 'm' and moving with velocity 'v'. If momentum increases by 20% then by how much kinetic energy will be increased?

As we discuss in relation between kinetic energy and momentum that,
twice of kinetic energy is equal to the product of momentum and velocity. that is given by,

2 K.E = PV

here momentum increased by 20% so that Kinetic energy is Increasing by 40%.

Relationship between momentum and kinetic energy equation

The relationship between momentum and kinetic energy equation says that, the kinetic energy of a body of mass 'm' moving with velocity 'v' is equal to the square of the momentum divided by mass of the body. Which is establish as:

K.E = p2 / 2m.

Hence, the equation is K.E = p2 / 2m.

Graph between kinetic energy and momentum when mass is constant

Here we plot a graph between kinetic energy and momentum when mass is constant. we can also plot graph when momentum is constant but here we take mass as constant.