### 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?

### Which Twin Is Older?

A simplified account of special relativity and the twins paradox.

### 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 Lorentz Force Law

##### Stage: 5 Challenge Level:

The Lorentz force law tells us how a non-relativistic charged particle $P$ moves under the influence of electric and magnetic forces:
$$m{\bf a(t)} = q\left({\bf E(t)}+{\bf v(t)} \times {\bf B(t)}\right)$$
In this vector differential equation, ${\bf a}$ and ${\bf v}$ are the acceleration and velocity of the particle of mass $m$ and charge $q$; The vectors ${\bf E}$ and ${\bf B}$ are the electric and magnetic fields respectively.

The motion of the particle in various electric and magnetic fields is described. In each case, what can we deduce about the vectors ${\bf E}$ and ${\bf B}$? Can you find particular solutions for fields giving rise to each motion? Describe clearly the configuration of the fields, along with an initial velocity, paying attention to whether various quantities are parallel or orthogonal to each other.

1) The particle sits at the origin and does not move.

2) The particle moves with a constant velocity, passing through the origin at time $t=0$.

3) The particle moves in a circle in the $x$-$y$ plane, centred on the origin.

4) The particle moves in a spiral up around the $z$-axis.

Once you have your solutions, could you see how to relate these back directly to the original differential equation?

Can you solve the equations for any other forms of ${\bf E}$ and ${\bf B}$?

Can you construct fields for which the kinetic energy of the particle will continually increase?