Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / Section A

For a photon, $E=pc$ and $p=h/\lambda$. Therefore

The de Broglie wavelength fixes the same momentum $p=h/\lambda$, but the nonrelativistic electron has $K=p^2/(2m_e)$:Here $K_e/(m_e{c}^2)\simeq 0.00118$, consistent with the stated nonrelativistic approximation. The photon energy and electron kinetic energy are not interchangeable.

For equal mean mass densities, $M_a/M_E=(r/R_E{)}^3$. Thus the asteroid's surface gravitational acceleration is $g_a=g_{E}r/R_E$ and its escape speed isAn explicit estimate of jumping ability is an Earth jump height $H=0.50\mathrm{m}$, giving $v_J=\sqrt{2g_{E}H}\simeq 3.1\mathrm{m}{\mathrm{s}}^{-1}$. ConsequentlyThis is an order-of-magnitude estimate, scaling with assumed take-off speed; no unique human jump speed is supplied. A spherical nonrotating asteroid and negligible atmosphere are assumed.

Convert both radii to metres before evaluating the moment of inertia:Its linear differential isFor independent standard uncertainties, add the three contributions in quadrature:If the quoted errors are instead bounds, their first-order absolute contributions add, giving $I\simeq (1.177\pm 0.042)\times 10^{-3}\mathrm{kg}{\mathrm{m}}^2$. Their statistical interpretation is not specified; the two conventions should remain distinct.

Spherical symmetry makes the electric field radial and constant in magnitude on each concentric Gaussian surface. The enclosed charge isGauss's law therefore givesThe direction is outward for $a>0$ and inward for $a<0$; the field vanishes at the centre. No exterior boundary condition changes this spherically symmetric interior result.

Write $K\propto m^{\alpha }{R}^{\beta }{h}^{\delta }$. Matching dimensions to $M{L}^2{T}^{-2}$ givesThe dimensionless factor $C$ cancels when the same confinement geometry and state are scaled. HenceUsing $m_p/m_e\simeq 1836$ gives this value. It remains small compared with the proton rest energy, so the nonrelativistic mass-radius scaling is self-consistent.

Use the pion rest energy supplied through its electron-mass ratio:Thus

The laboratory mean lifetime is dilated to $\gamma {\tau }_0=44.7\mathrm{ns}$. Multiplying by the laboratory speed givesThis is the mean of the exponential decay-distance distribution, not a fixed distance travelled by every pion.