Quick questions on Length contraction and relativistic momentum (QCE Physics Unit 4)

Length contraction acts only along the direction of motion. A rod's transverse dimensions (perpendicular to its motion) are unaffected. A passing ball would be seen as flattened along its direction of motion but unchanged in the other two dimensions.

The proper length is the longer length, measured in the frame where the rod is at rest. Other observers measure a shorter length.

Like time dilation, length contraction is not optical illusion. The atmosphere appears thinner to a relativistic muon (in the muon's frame, the atmosphere is contracted; in Earth's frame, time is dilated; both descriptions agree on whether the muon survives to the surface).

At low speeds, $\gamma \approx 1$ and $p \approx m v$ (classical). At high speeds, $\gamma$ grows without bound as $v \to c$. The momentum required to push a particle to a speed approaching $c$ grows without limit; no particle with rest mass can reach $c$.

Designing a particle accelerator requires the relativistic momentum formula. Synchrotrons (where charged particles travel in a circle in a magnetic field) must use $p = \gamma m v$ when calculating the magnetic field needed to keep particles in their circular path.

Some high-energy cosmic ray particles have $\gamma > 10^{10}$ (extraordinarily relativistic). The classical momentum formula is hopelessly wrong; only $p = \gamma m v$ works.

The relativistic energy $E = \gamma m c^2$ (covered in the mass-energy dot point) diverges as $v \to c$. Infinite energy would be required.

Photons have zero rest mass and travel at $c$. They carry momentum $p = E/c$ (consistent with $p = \gamma m v$ in an appropriate limit).

Charged particles in accelerators (LHC, Tevatron, etc.) follow trajectories that match relativistic momentum predictions. Without the $\gamma$ correction, the accelerator's magnetic systems would not bend the particles correctly.

High-energy pions in cosmic-ray cascades have lifetimes consistent with time dilation, and trajectories consistent with relativistic momentum.

Proper length is measured in the rest frame of the object (the longer length). Contracted length is measured in any other frame.

At $v / c > 0.1$, the relativistic correction is measurable; above $v / c > 0.5$, it is dominant. Use $p = \gamma m v$ whenever in doubt.

State the length-contraction formula and define proper length. [2 marks]

Calculate the contracted length of a $30 \text{ m}$ rod at $v = 0.80c$. [3 marks]

A cosmic-ray muon at $0.99c$ tracked over a $10 \text{ km}$ atmospheric column. (a) Calculate $\gamma$ and the contracted length in the muon's frame. (b) Calculate the relativistic momentum of a $1.88 \times 10^{-28} \text{ kg}$ muon.