A bus drives along a straight road at a constant speed. You can run with a speed two times less than the speed of the bus. You first notice the bus at a point A. From what region around the street can you reach the bus?
Some explanation of this direct translation is in order. "with a speed two times less than the speed of" means "half as fast as". You may assume that the bus will stop (instantaneously) and pick you up wherever along the road you reach the bus. Because you can't run as fast as the bus, you can't reach it from behind. You can certainly reach it from in front of point A -- you can just stand and wait for it, or you can run towards it and meet it closer to point A. You can also catch it if you are some distance in front of point A, and some distance away from the road, because you will have enough time to run to the road before the bus gets there. Thus there is some region around the road and in front of A from which you can catch the bus. We want to know the boundary of that region.
The geometry of this problem is similar to that of sonic shock waves. In your solution you might explain why.
A massive plate lies on two parallel, massless cylinders of different radii, making an angle α with the horizontal. The cylinders roll on a horizontal plane. There is no slipping. Find the acceleration of the plate.