Relativistic Mass Formula

Last Updated : 04 Feb, 2024

In physics, the mass idea discourse is quite common. When there are comparisons of length and time in distinct frames, the well-known special theory of relativity says a lot more about relativistic mass. When the body is moving, the relative change in mass is also experienced. This is referred to as relativistic mass. When an item is moving, it experiences mass increase, which is similar to length contraction and time dilation.

Relativistic Mass

The mass of an object that varies with its own speed as the said object approaches the speed of light is called its relativistic mass. It rises with speed and approaches infinity as the speed rises to the speed of light. Its unit of measurement is meters per second (m/s) and the dimensional formula is given by [M¹ L⁰ T⁰].

In particle and nuclear physics, the phrase “relativistic mass” is rarely used, and special relativity writers prefer to refer to the relativistic energy of the object instead.

Example

The weight of the particle accelerator + electrons system can be raised by the translational weight of an electron, not through the electron’s resting mass, if an electron in a cyclotron moves in circles with a translational velocity.

Formula

m = $\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

where,

m_o denotes the rest mass of the object
v denotes the velocity of the moving body
c denotes the velocity of light

Sample Problems

Problem 1: A 10 kg object travels in the air at a velocity of 0.77 c. Calculate its rest mass.

Solution:

Given: m = 10 kg, v = 0.77 c and c = 3 × 10⁸ m/s.

Since, m = $\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

$10=\frac{m_0}{\sqrt{1-\frac{(0.77)^2c^2}{c^2}}}$

10 = m₀/0.6830

m_o= 6.3 kg

Problem 2: A 20 kg object travels in the air at a velocity of 0.67 c. Calculate its rest mass.

Solution:

Given: m = 20 kg, v = 0.67 c and c = 3 × 10⁸m/s.

Since, $m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

$20=\frac{m_0}{\sqrt{1-\frac{(0.67)^2c^2}{c^2}}}$

20 = m₀/0.7423

m_o = 14.8 kg

Problem 3: A 10 kg object travels in the air at a velocity of 0.99 c. Calculate its rest mass.

Solution:

Given: m = 10 kg, v = 0.99 c and c = 3 × 10⁸ m/s.

Since, $m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

$10=\frac{m_0}{\sqrt{1-\frac{(0.99)^2c^2}{c^2}}}$

10 = m₀/0.1410

m_o = 1.41 kg

Problem 4: A 10 kg object travels in the air at a velocity of 0.43 c. Calculate its rest mass.

Solution:

Given: m = 10 kg, v = 0.43 c and c = 3 × 10⁸ m/s.

Since, $m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

$10=\frac{m_0}{\sqrt{1-\frac{(0.43)^2c^2}{c^2}}}$

10 = m₀/0.9028

m_o = 18.056 kg

Problem 5: A 10 kg object travels in the air at a velocity of 0.33 c. Calculate its rest mass.

Solution:

Given: m = 10 kg, v = 0.33 c and c = 3 × 10⁸ m/s.

Since, $m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

$10=\frac{m_0}{\sqrt{1-\frac{(0.33)^2c^2}{c^2}}}$

10 = m_o/0.9439

m_o = 9.4 kg

Problem 6: Calculate the mass of a particle at a velocity of 0.21 c if its rest mass is 10 kg.

Solution:

Given: m_o = 10 kg, v = 0.21 c and c = 3 × 10⁸ m/s.

Since,

$m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

$m=\frac{10}{\sqrt{1-\frac{(0.21)^2c^2}{c^2}}}$

m = 10.22 kg

Problem 7: Calculate the mass of a particle at a velocity of 0.45 c if its rest mass is 20 kg.

Solution:

Given: m_o = 20 kg, v = 0.21 c and c = 3 × 10⁸ m/s.

Since,

$m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

$m=\frac{20}{\sqrt{1-\frac{(0.45)^2c^2}{c^2}}}$

m = 22.39 kg