Question

The velocity of an electron, whose mass is ten times its rest mass, is 
1. 3.3 × 10⁸ m/sec
2. 2.98 × 10⁸ m/sec
3. 4.2 × 10⁷ m/sec
4. ten times the velocity of light

Answer 1

Correct Answer - Option 2 : 2.98 × 10⁸ m/sec

CONCEPT:

• The mass of an object measured when it is at rest (v = 0) is known as its rest mass. The rest mass is denoted by m_o.
• According to Einstein's special theory of relativity, the mass of an object moving with a velocity comparable to the speed of light appears more than its rest mass.
• As an object moves with relativistic velocities (v ~ c), where c = 3 x 10⁸ m/s is the velocity of light, its mass modifies as 

\(\Rightarrow m=\frac{m_o}{\sqrt{1-\frac{v^2}{c^2}}}\)

• So, the mass of an object is a function of the velocity of the object. 

CALCULATION:

• In the question, the mass of the electron is 10 times its rest mass.

\(⇒ m=10m_0 \\ ⇒ \frac{m_o}{\sqrt{1-\frac{v^2}{c^2}}}=10m_o\\ ⇒ 1-\frac{v^2}{c^2}=0.01\\ ⇒ \frac{v^2}{c^2}=0.99\\ ⇒ v=0.99 c= 2.98 \times 10^8 m/s \)​

• Therefore, option 2 is correct.

• Einstein's special theory of relativity is valid only for objects moving with relativistic velocities (v~c).
• In the limit v << c, the theory yields classical results.

\(v<<c \Rightarrow \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\approx 1\\ \Rightarrow m=m_o\)