Question 13 of 17intermediate🔍 AnalyzeShort Answer2 marks

A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ m₀ of a particle in terms of its speed v and the speed of light, c. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes: m = m₀ / (1 - v²)^(1/2). Guess where to put the missing c.

Correct Answer

The missing c should be placed in the denominator inside the bracket. The correct relation is:

m=m0(1−v2c2)1/2m = \frac{m_0}{\left(1 - \frac{v^2}{c^2}\right)^{1/2}}

Reasoning: According to the principle of homogeneity of dimensions, only quantities with the same dimensions can be added or subtracted. In the boy's expression (1−v2)1/2(1 - v^2)^{1/2}, we are subtracting v2v^2 (which has dimensions [L2T−2][L^2 T^{-2}]) from 1 (which is dimensionless). This is dimensionally incorrect.

Since c (speed of light) has the same dimensions as velocity [LT−1][L T^{-1}], the ratio v/cv/c becomes dimensionless. Therefore, v2/c2v^2/c^2 is also dimensionless and can be subtracted from 1. This makes the equation dimensionally correct.

Exercise: EXERCISES | Q: 1.13 | (Page 11)
For More Understanding

Explanation

The textbook context emphasizes using dimensional analysis to check equation correctness. The principle of homogeneity states that only quantities with same dimensions can be added/subtracted. In (1-v²), v² has dimensions [L²T⁻²] but 1 is dimensionless - violating this principle. Since c has dimensions [LT⁻¹], dividing v² by c² makes it dimensionless, allowing valid subtraction from 1.

Solution Steps

  1. Step 1: Identify that (1 - v²) is dimensionally incorrect since v² has dimensions [L²T⁻²] while 1 is dimensionless.

  2. Step 2: Apply the principle that only quantities with same dimensions can be subtracted.

  3. Step 3: Recognize that c (speed of light) has dimensions [LT⁻¹], same as velocity.

  4. Step 4: Make v² dimensionless by dividing by c², giving the correct form: m = m₀/(1 - v²/c²)^(1/2)

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