Problems

The Norman army is advancing with constant speed \(u\) towards the Saxon army, which is at rest. When the armies are \(d\) apart, a Saxon horseman rides from the Saxon army directly towards the Norman army at constant speed \(x\). Simultaneously a Norman horseman rides from the Norman army directly towards the Saxon army at constant speed \(y\), where $y > u$. The horsemen ride their horses so that \(y - 2x < u < 2y - x\). When each horseman reaches the opposing army, he immediately rides straight back to his own army without changing his speed. Represent this information on a displacement-time graph, and show that the two horsemen pass each other at distances \[ \frac{xd }{ x + y} \;\; \mbox{and} \;\; \frac{xd(2y -x-u)} {(u+x ) ( x + y )} \] from the Saxon army. Explain briefly what will happen in the cases (i) \(u > 2y - x\) and (ii) \(u < y - 2x\).