Determine the ratio of the relativistic kinetic energy to th

Determine the ratio of the relativistic kinetic energy to the nonrelativistic kinetic energy (1/2mv²) when a particle has a speed of (a) 1.66 × 10^-3c. and (b) 0.951c.

Solution

Relativistic kinetic energy is mc²{[1/sqrt(1 - ²) ] – 1},
nonrelativistic kinetic energy is m v²/2 = 0.5mv²

(a)

= v / c= 1.66 * 10^-3*c / c = 1.66 * 10^-3

1/(1-²) =1.00000138

1/(1-²)] – 1 = 0.00000138 = 1.38 * 10^-6

Relativistic kinetic energy is 1.38 * 10^-6 *mc²

The ratio is 1.38 *10^-6 * mc² / [0.5 *m * (1.66 * 10^-3)² * c² ]=

= 1.38 * 10^-6 / 1.3778 * 10^-6 = 1.001596

(b)

= v / c= 0.951 * c / c = 0.951

1/(1 - ²) = 3.234,

1/(1-²)] – 1 =2.234

Relativistic kinetic energy is 2.234* mc²

The ratio is 2.234 * mc² / [0.5 *m *( 0.951)² * c² ] = 4.94