1. (The clepsydra, or water clock) A 12 hour water clock is to be designed with the dimensions shaped like the surface obtained by revolving the curve y = f(x) around the y-axis. What should be this curve, and what should be the radius of the circular bottom hole, in order that the water level will fall at the constant rate of 4 inches per hour?
Height = 4 ft
Radius of the top of the clock = 1 ft
Hint: A(y) dy/dt = - k* sqrt(y)
2. Arthur Clarke's The Wind from the Sun describes Diana, a spacecraft propelled by the solar wind. Its aluminized sail provides it with a constant acceleration of 0.0001 g = 0.0098 m/s2. Suppose this spacecraft starts from rest at time t = 0 and simultaneously fires a projectile (straight ahead in the same direction) that travels at one-tenth of the speed c = 3 x 108 m/s of light. How long will it take the spacecraft to catch up with the projectile, and how far will it have traveled by then?