The head start

A tortoise and a hare agreed to race along a path 120 metres long. The tortoise walks two metres a minute and the hare runs six metres a minute.

Track $120$ m. Tortoise: $2$ m/min. Hare: $6$ m/min.

Track $L=120$ m; $v_T=2$, $v_H=6$ m/min.

The hare was fair: she gave the tortoise a head start of sixty metres. So the tortoise began halfway along the path, and the hare began at the very start.

Head start for the tortoise: $60$ m. The tortoise starts at $60$ m, the hare at $0$.

$s_T(0)=60,s_H(0)=0$.

The hare worked it out. Running without stopping, she needs 120 divided by 6, which is twenty minutes. The tortoise has sixty metres to go at two metres a minute: thirty minutes. "I have ten minutes to spare," she said, and lay down for a nap of fifteen minutes.

Hare, no stopping: $120÷6=20$ min. Tortoise: $(120-60)÷2=30$ min. Her spare time is $30-20=10$ min, and she naps $15$ min.

$t_H=120/6=20$, $t_T=(120-60)/2=30$, $30-20=10$; nap $n=15$.

The tortoise never stopped, and arrived after thirty minutes. The hare woke and ran: twenty minutes of running and fifteen of sleep make thirty-five in all. A nap longer than your spare time turns a win into a loss.

The tortoise arrives at minute $30$. The hare takes $20+15=35$ min, and $35>30$. A nap longer than $10$ min turns a win into a loss.

$n=15>10\Rightarrow t_H+n=35>30=t_T$.