# Relativistic Mass Formula

The mass concept dialogue is a very standard one in physics. The well known special theory of relativity voices a lot more about relativistic mass when there are comparative measurements of length and time in dissimilar frames. The relative change in mass is also perceived when the body is in motion. This concept is relativistic mass. Comparable to length contraction and time dilation a thing called mass increase happens when the object is in motion.

## Formula of Relativistic Mass

The relativistic mass formula is articulated as,

$$\begin{array}{l}m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\end{array}$$

Where,

• the rest mass is mo,
• the velocity of the moving body is v,
• the velocity of light is

### Solved Examples

Problem 1: An object in motion has a mass of 12 kg and travels in the air with a velocity 0.82. What would be its rest mass?

Known:

(Mass) m = 12 kg,

(Velocity) v = 0.82c,

c = 3 × 1088 m/s22.

The relativistic mass formula is articulated as,

$$\begin{array}{l}m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\end{array}$$
$$\begin{array}{l}12=\frac{m_{0}}{\sqrt{1-\frac{(0.82)^{2}c^{2}}{c^{2}}}}\end{array}$$

Rest mass, mo = 7.2 kg (approximately)

Thus, the rest mass of the given object is 7.2 kg.

Problem 2: A particle of mass 1.67 × 10−24 kg travels with velocity 0.65. Compute its rest mass?

Given: Mass m = 1.67 × 10−24 kg, v = 0.65c, c = 3 × 108 m/s2.

The relativistic mass formula is articulated as,

$$\begin{array}{l}m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\end{array}$$
$$\begin{array}{l}1.67\times10^{-24}=\frac{m_{0}}{\sqrt{1-\frac{(0.65)^{2}c^{2}}{c^{2}}}}\end{array}$$

Rest mass, mo = 1.26 × 10−24 kg (approximately).

Thus, the rest mass of the particle is 1.26 × 10−24 kg.

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