# Variation of Mass with Velocity Notes for Engineering Physics

Variation of Mass with Velocity Notes for Engineering Physics:

According to Einstein, the mass of the body in motion is different from the mass of the body at rest. We consider two inertial frames S and S’ as in Figure

We now consider the collision of two bodies in S’ and view it from the S. Let the two particles of masses m_{1} and m_{2} are travelling with velocity u ‘ and-u ‘ parallel to x-axis in S’. The two bodies collide and after collision they coalesced into one body.

**In System S: Before Collision:**

Mass of bodies are m1 and m2• Let the their velocities are u_{1 }and u_{2} respectively.

**In System S: After Collision**:

Mass of the coalesced body is (m_{1}+ m_{2}) and the velocity Is v .

Using law of addition of velocities;

Applying the principle of conservation of momentum of the system before and after the

collision, we have,

m_{1} u_{1} +m_{2} u_{2} = (m_{1} +m_{2})v

Now, using equations (1) and (2), we have

M_{1}/m_{2 }= [√ 1-(u_{2} /c)^{2} /√ 1-(u_{1} /c)^{2} ]

Let the body of mass m_{2} is moving with zero velocity in S before collision, i.e., u_{2} = 0,

hence, using equation (3), we have,

m_{1 }/m_{2 = }1 / √ 1-(u_{1}/c)^{2}

Using common notation as m_{1}= m, m_{2} = m _{0} , u_{1} = v, we have by using equation (4).

This is the relativistic formula for variation of mass with velocity, where m _{0} is the rest mass and m is the relativistic mass of the body. There are a large numbers of experimental observations of this enhancement of mass of particles in high energy physics

**I. When v << c**

**v ^{2} << c^{2}, v ^{2} / c_{2} is negligible as compared to 1 => c m =m_{0}**

When velocity of the moving particle is much smaller as compared to velocity of light,

relativistic mass equals the rest mass.

**II. When v= c**

** V ^{2} =c^{2} ,v^{2} /c^{2} =1 => [1- v^{2} /c^{2} ] ,< 1 => m >m_{0}**

When velocity of the moving particle is comparable to velocity of light, relativistic mass of the body appears to be greater than the rest mass.

**III. When v = c**

**V ^{2} =c^{2} , v^{2} /c^{2} =1 => m**

When velocity of the moving particle is exactly equal to velocity of light, relativistic mass of the body appears to be infinite and this is an impractical concept.

**IV. When v > c**

**V ^{2} > > c^{2} ,v^{2} /c^{2} > 0 m = Imaginary**

When velocity of the moving particle is greater as compared to velocity of light., relativistic mass becomes imaginary and this is an impractical concept.