In each of these missions, a civilian informs Cole of the Dust Men driving a large bus rigged with assault weapons and armor plating, using it to cause mass panic and murder by continually firing at innocent pedestrians in sight. Cole must sabotage it before many civilians end up seriously injured or dead. The only way he can destroy the bus is by getting on top of it and overloading its generator with his Lightning Bolts.
You should not take a bus head on unless you attempt the tactic mentioned above, as it is armored with corrugated pieces of iron, and there are heavy machine guns on all sides of it being fired by Dust Men inside. Also, you cannot climb onto the roof, as there is barbwire around the unprotected roof exploiting the generator.
If you plan to jump on the roof of the bus from the roof of the building, think ahead. You won't know how successful you are until you jump. Here's an easy way to tell how you should time your jumps; the higher the elevation, the quicker you should jump. For more precision, use Static Thrusters.
The only way you'll ever know if the bus is out of commission is if a cutscene is activated, showcasing the destruction of the bus.
If the bus blows up, Cole will take no damage at all.