How to find rest energy and kinetic energy of a ball?

Question

Calculate the rest energy and the kinetic energy of:
a) A 100g ball travelling with the speed of 100m/s
b) An electron travelling at a speed of 0.999c.
Which one has greater rest energy? Which one has greater Kinetic energy? Don’t forget you are dealing with relativistic effects in at least one case.

Answer 1

The rest mass of an electron is about 9.109×10⁻³¹ kilograms, clearly many orders of magnitude less than 100g. The rest energy is given by e=mc^2, hence the 100 gram ball wins by a vast margin.

For kinetic energy, the speed of the ball is given as 100 m/s, which is so far below the speed of light (299792458 m/s) that a non-relativistic calculation can be made: e = 1/2 m*v^2 = 1/2 * (100g) * (100 m/s)^2 = 1/2*(0.1 kg)*(100 m/s)^2) = 500 kg*(m/s)^2 = 500 joules.

Now, for an electron moving at 0.999C, relativistic effects should be considered. The amount of energy that needs to be applied to an electron to accelerate it to a velocity of 0.999C is the rest energy multiplied by the factor 1/(1-(v/c)^2)^(1/2) = approximately 22. So, the equivalent mass of an electron at 0.999C is only about 23 times its rest mass, still far lower than the 100 gram ball.

In order for an electron to have a relativistic mass of 100 grams, you would need to solve for "v" in the equation 0.1 kilogram = (9.109×10⁻³¹ kilograms)*(1 + 1/(1-(v/c)^2)^(1/2)). Actually, it would be better to solve for "v/c" to a very high precision. I leave this as an exercise to the readers, What answers do you get (use scientific notation)?