Question

What fraction of light speed does each of the following speeds represent? If any calculation is required, use our usual Newtonian methods and ignore any relativistic effects. Please be sure to comment on whether the Newtonian results are a good approximation to the correct speed.

a. A billiard ball moving at 1 m/s.

b. A major league fastball crossing the plate at 90 mph.

c. A satellite orbiting Earth in low Earth orbit (say 100 km). Earth radius is 6.4 x 106 m.

d. A proton dropped from infinitely far away from a white dwarf star of mass 1.4 solar masses and radius 5000 km. Calculate its kinetic energy when it crashes into the surface of the star and use that to find its speed. (Hint: mgh is not a good way to write gravitation PE; instead, use -GM1m2 / r.)

e. A spaceship that starts from rest and accelerates at 1 m/s2 for 50 years.

f. An electron in the circular LEP accelerator at CERN in Geneva. This is the highest energy accelerator in the world, and electrons are accelerated to energies of 1.01 x 1011 electron-volts. (1 Joule is 1.6 x 10-19 electron-volts.)

Answer 1

The speed of light c = 3 x 10⁸ m/s

a) v = 1 m/s.

Then fraction f = v / c = 1/(3 x 10⁸) = 3.33 x 10^-9.

b) v = 90 mph = 144.81 kmph = 40.23 m/s

Then fraction f = v/c = 40.23/(3 x 10⁸) = 1.34 x 10^-7.

c) Velocity of the satellite is given by

Then fraction f = v/c = (7.9 x 10³)/(3 x 10⁸) = 2.63 x 10^-5.

d) Mass of proton Mp = 1.67 x 10^-27 kg.

KE = G Ms Mp / r = 6.67 x 10^-11 x 1.4 x 1.99 x 10³⁰ x 1.67 x 10^-27/ (5000 x 10³) = 6.21 x 10^-14 J.

Then KE = 1/2 Mp v² => v = sqrt(2 KE / Mp) = sqrt(2 x 6.21 x 10^-14 / 1.67 x 10^-27) = 8.62 x 10⁶ m/s

Then fraction f = v/c = (8.62 x 10⁶)/(3 x 10⁸) = 2.87 x 10^-2.