Is it permissible to perform the Fajr prayer in the bus while going to work, if its time begins after departure and stopping is not possible, and is it permissible to perform the prayer when sleepiness overcomes one in it?

Prayer must be performed at its prescribed times. Whoever knows that travel will cause him to miss it, then he must perform it before boarding, or hasten to board so that he can perform it immediately upon disembarking. If the prayer is one that can be combined (like Dhuhr and Asr, Maghrib and Isha), then it is permissible to combine them by performing them earlier (jama' taqdeem) or later (jama' ta'kheer) to avoid praying in the vehicle and the deficiencies that result from it.

Fajr prayer cannot be combined with any other prayer. So, if a Muslim is compelled to pray in the vehicle because it cannot be stopped, and fears missing the time (after sunrise), he should pray while riding, performing as many of its pillars and conditions as he is able. That which he is unable to do is waived for him, and it is not permissible to delay it beyond its time. He should pray standing if possible and face the Qibla (definition). If he cannot perform bowing (ruku') and prostration (sujood), he should lean for both and make the prostration lower than the bowing. It is preferable to lean for bowing while standing if possible, then sit and lean for prostration.

As for deep sleep during which a person is unaware of (bodily) occurrences, it invalidates ablution (wudu'), unlike light sleep. If ablution is invalidated and performing wudu' is impossible on the bus, he should perform tayammum (definition) by striking his hands on the floor of the bus and wiping his face and palms, then pray, and he should not delay the prayer beyond its time.