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 m₁ and m₂ 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 u₁ and u₂ respectively.

In System S: After Collision:

Mass of the coalesced body is (m₁+ m₂) 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,

m₁ u₁ +m₂ u₂ = (m₁ +m₂)v

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

M₁/m₂ = [√ 1-(u₂ /c)² /√ 1-(u₁ /c)² ]

Let the body of mass m₂ is moving with zero velocity in S before collision, i.e., u₂ = 0,

hence, using equation (3), we have,

m₁ /m_{2 =} 1 / √ 1-(u₁/c)²

Using common notation as m₁= m, m₂ = m ₀ , u₁ = 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 

v² << c², v ²  / c₂ is negligible as compared to 1 => c m =m₀

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

 V² =c² ,v² /c² =1 => [1- v² /c² ] ,< 1 => m >m₀

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

V² =c² , v² /c² =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

V² > > c² ,v² /c² > 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.

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