Question

A. At what speed does the classical momentum, p=mv, give an error, when compared with the relativistic momentum, of 1.28 %

B. At what speed does the classical momentum, p=mv, give an error, when compared with the relativistic momentum, of 4.76 % ?

Answer 1

Let p_c be the classical momentum and p_r be the relativistic momentum.

Consider the following ratio.

   f = p_r- p_c /p_r

     = 1 - (p_c /p_r)

Substitute the values.

     = 1 - m v / (m v / ?1 -(v² / c²))

By simplifying.

     = 1 - ?1 - (v² /c²)

   ?1 - (v² /c²) = 1 - f

   1 - (v² / c²) = (1- f)²

   v = c ?1 - (1 - f)²

(a) if f = 0.0128

   v = c ?1 - (1 -0.0128)²

      = 0.159 c

(b) if f = 0.0476

   v = c ?1 - (1 -0.0476)²

      = 0.3048c