### Lunar Leaper

Gravity on the Moon is about 1/6th that on the Earth. A pole-vaulter 2 metres tall can clear a 5 metres pole on the Earth. How high a pole could he clear on the Moon?

### High Jumping

How high can a high jumper jump? How can a high jumper jump higher without jumping higher? Read on...

### Whoosh

A ball whooshes down a slide and hits another ball which flies off the slide horizontally as a projectile. How far does it go?

# The Ultra Particle

##### Stage: 5 Challenge Level:

Note: Assume standard Newtonian mechanics for the first parts of this problem. Required data are provided at the foot of the problem.

On October 15, 1991 an ultra-high energy proton came to earth, near Utah, from space with an energy of $3.2\pm 0.9\times 10^{20}$ electron volts. This REALLY is a lot of energy for a proton to possess, so much so it was dubbed the 'Oh-my-god' particle! To see why, explore the energy it contains by comparing this figure with the kinetic energy for the motion of more every-day objects - find an everyday situation which really gives an intuitive sense of the amount of energy of this ultra-high energy proton.

If you used the Newtonian expression $KE=\frac{1}{2}mv^2$ for the energy, how fast would the proton be travelling? How does this compare with the speed of light? What does this tell you?

Imagine that a small, hand-sized, ball of meteoric iron with the same kinetic energy per kilogram as this ultra-high energy proton struck earth. Analyse the possible effect this would have.

Extension: Clearly, the energies referred to in the question push the furthest reaches of Einstein's theory of special relativity. Use this situation to determine the actual velocity of the proton relative to Earth using the formula involving the rest mass $m_0$ of a stationary object and the speed of light $c$
$$E= \frac{m_0c}{\sqrt{1-\frac{v^2}{c^2}}}\;.$$

Data for the problem

1 electron volt $\left(\mathrm{eV}\right)$ = $1.602 176 46 \times 10^{-19}\textrm{ J}$
Rest mass of the proton is $9.3828\times 10^8\textrm{ eV}$
Mass of proton $1.672 621 58\times 10^{-27}\textrm{ kg}$
Speed of light $2.99 792 458\times 10^8\textrm{ ms}^{-1}$

NOTES AND BACKGROUND

Cosmic rays are simply particles of matter moving through space; typically high energy protons, helium nuclei or electrons. Modern models of physics currently under investigation by theoretical physicists predict bounds on the energies of cosmic rays or the appearence of more exotic particles in cosmic rays.

Ultra-high energy particles are occasionally found in cosmic rays. These defy well understood bounds, confound theoretical physicists and help to drive forward the bounds of our understanding of the universe. An interesting article on the particle referred to in this problem is found here.