A crane lifts a mass $m$ to the 10$^{th}$ floor of a building in a certain time. It then raises the same mass to the 20$^{th}$ floor of the same building in double the time. How much more energy and power are required? Assume that the height of all floors is equal.

Let the height of each floor be $h$.

Energy required to lift a mass is given by:

$$\text{Potential Energy} = mgh$$

For the 10th floor:

$$E_1 = mg(10h)$$

For the 20th floor:

$$E_2 = mg(20h)$$

$$E_2 = 2E_1$$

Therefore, twice the energy is required to lift the mass to the 20th floor.

Power is given by:

$$P = \frac{W}{t}$$

If time taken for the 10th floor is $t$, then time taken for the 20th floor is $2t$.

$$P_1 = \frac{10mgh}{t}$$

$$P_2 = \frac{20mgh}{2t}$$

$$P_2 = \frac{10mgh}{t} = P_1$$

Therefore, the power required remains the same.