Bike

Suppose that a drop detaches from the wheel when it is at angle $\alpha$ with the vertical. The bike is moving horizontally with the velocity $v$ relative to the ground and the points of the wheel is rotating with the velocity $v$ tangentially relative to the wheel. You need to add these velocities vectorially if you want to find the velocity of the drop relative to the ground. Write the equations of motion for the drop in horizontal and vertical directions. Find the expression for $y_{max}$ for an arbitrary angle $\alpha$ and then differentiate to find the angle for which drop reaches its maximum height.