Two math questions for bikers: with interesting solutions!

1. When you are biking and you stop for 1 minute to put on a coat and your bike partner goes on, how long will it take before you keep up, by going 2 km/h faster? After what distance do you meet again?
Solution (simplified, ignoring the acceleration phase):
Between begin of stop and re-unite person 1 and person 2 have traveled the same distance (D). During the stop phase (Tstop) person 2 has continued to cycle with velocity 2 (V2). After the stop phase, person 1 has to gain with velocity 1 (V1) the distance that person 2 has cycled during the stop phase. The time required (T) to keep up is hence given by:
T = Tstop*V2 / V2-V1
and the distance:
D = V1*T
So say that the velocity of person 2 is 20 km/h and that of person 2 is 22 km/h to keep up. Tstop is one minute, 1/60 hour. If you use the formula you find that the time needed to keep up is 10 minutes and the distance 3.7 km.
Of course, numbers increase when the base velocity of the trip (and hence person 2) is higher or the time stopped is longer. Every second counts! Better you stop both :-)

2. When you are bicycling in the rain, would it help to go faster?
You are earlier at your endpoint by reducing the total time in the rain, but you also get wetter because you get more raindrops per time unit. By biking faster, do you get wetter, are you getting less wet, or doesn't it matter at all? Does the same apply for walking and motorcycling?
Rain is usually measured as rate in mm/h. After making some calculations on a sketch paper I found out that this problem is independent of surface and distance! The rain rate is normally applied to a non-moving object. The rain "receive rate" depends on the velocity of the object, not on it's surface. An object that moves twice as fast as another, receives twice as much rain per time unit. The rain rate can just be multiplied by the velocity to get the rain receive rate. If you then multiply the rain receive rate with the time required for a fixed distance, you will find that two objects moving with different speeds have received the same amount of rain at the end of the track. So when you are going faster, you are only faster to get dry clothes in a warm house! But if you know the rain is going to stop before you get home, slow down!

Feel free to comment if you have any addition or correction!