relativity6

B. Tech Physics

Chapter 1

Theory of Relativity

Lecture 1.6

Concept of Relativistic Momentum/Variation of mass

with velocity:

The conservation of linear momentum states that when two

bodies collide, the total momentum remains constant,

assuming the bodies are isolated.

If the velocities of the colliding bodies are calculated in a

second inertial frame S’ using the Lorentz

transformation, and the classical definition of momentum

p = mu

applied, one finds that momentum is not

conserved in the second reference frame.

However, because the laws of physics are the same in all

inertial frames, and that should also be for law of

We consider a collision between two particles of equal mass.

In frame S, we assume the particles approach each other at

Speed v and after collison, stick to each other and come

to rest.

Fig (b) for frame S’

Now, we view the collision from frame S’ which is

Moving with velocity v along x-axis.

Let the velocities observed from frame S’ are v

’,

v

’ before collision.

The velocities observed from S’are,

Now, applying momentum conservation in frame S’, we get

However, if we modify the definition of momentum to,

Then is can be shown that the momentum will be conserved

in both S and S’.

Note that some time rest mass is also denoted by m

and

is related to the mass of object in motion by relation,

Relativistic force : 2nd law in relativistic form:

However when then we have γ much less then one and we get the classical formula.

Einstein mass energy relation: The relation E = mc2 is known as

Einstein mass-energy relation. It says that the mass can be converted into energy and vice-versa.

Proof: The work done, W on an object by a constant force F because of which the object moves through distance s in direction of force is given by,

W = Fs ---(1)

If no other force acts on the object and object starts from the rest then the whole work done converts into K.E. i.e. KE = Fs.

In general when force is not constant then the K.E. is,

In the non-relativistic case the Kinetic energy of an object of mass m and speed v is (mv 2/2).

To get the relativistic form of the K.E. We shall use the relativistic Form of 2nd law of motion i.e.

---(3) Integrating by part

Above eq. Says that the kinetic energy of an object is

equal to the increase in its mass due to its relativistic

motion multiplied by the square of the speed of the

light.

Above Einstein mass-energy relation ship says that the mass can be created or destroyed but when this happens an equivalent amount Of energy simultaneously vanishes or comes into being and

Vice-versa.

Exercise: Consider the collision between two bodies of mass

m such that after collision total mass become M. For detail

Exercise: For the collision discussed in the section on

relativistic momentum, show that the momentum will

be conserved in both frame S and S’, if we use the

relativistic definition of momentum

Hint:

In your calculations use

M

as derived in last exercise

Relativistic energy-momentum relationship:

Consider an object of relativistic mass m and moving with velocity v. Its momentum is

---(1)

The relativistic mass is given by

---(2)

Last eq. further can be written as

---(3)

Also we know Einstein mass-energy relation is

---(4) Using (3) in (4) we have

Concept of massless particles:

The momentum of the particle is defined by

And energy is

E =

Now if m₀ = 0 and v<c, then E = p = 0.

However, when m₀ = 0 and v = c, than E = 0/0 and p = 0/0, which mean the energy and momentum are indeterminate and can possess any values. Thus a mass-less particle with finite value of E and p can exist provided they move with the speed of light.

Example : Photons are mass-less particle which move with speed c and have rest mass zero.

The total energy of such particles (using and m₀ = 0),

Energy, E = pc

Exercise: