In this problem, we are comparing the power requirements of two drones, where one is an exact replica of the other but with doubled linear dimensions. We need to find the relationship between the mechanical power input of the original drone, $P_1$, and the mechanical power input of the scaled-up drone, $P_2$.

Step-by-step explanation:

1. Scaling of dimensions:
   When the linear dimensions are doubled, every dimension (height, width, length) of the replica is twice that of the original. This affects certain physical properties such as volume, surface area, and mass.

2. Effect on volume and mass: The volume $V$ of the drone scales with the cube of the linear dimensions. Since the dimensions are doubled: $V_2 = 2^3 \times V_1 = 8 \times V_1$ Assuming the drone is made of the same material, the mass $m$ will also scale proportionally to the volume, meaning the mass of the new drone is 8 times that of the original: $m_2 = 8 \times m_1$

3. Effect on power requirement:
   The mechanical power input $P$ for a drone to hover is related to its mass and the gravitational force it needs to counteract to stay in the air. The power required is also dependent on the aerodynamic properties of the drone.

   The power to hover is proportional to the mass and the speed at which air must be displaced by the rotors. From aerodynamics, power scales with mass to the power of $3/2$. Therefore, the power requirement for the scaled-up drone, $P_2$, compared to the original, $P_1$, can be expressed as: $P_2 = P_1 \times \left(\frac{m_2}{m_1}\right)^{3/2}$

4. Substitute the mass ratio: We know that $m_2 = 8 \times m_1$, so: $P_2 = P_1 \times (8)^{3/2}$ $P_2 = P_1 \times 8^{1.5} = P_1 \times 22.627$

Thus, the new power requirement $P_2$ is approximately 22.6 times greater than the original power $P_1$.

Conclusion:

If the linear dimensions of the drone are doubled, the new power input $P_2$ will be approximately 22.6 times the original power input $P_1$.

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Would you like more details on how the power scaling works or clarification on any part of the solution?

Here are some related questions to deepen your understanding:

1. How would the power input change if the drone’s dimensions were tripled instead of doubled?
2. What other factors, besides mass, might affect the power required for a drone to hover?
3. How is the power-to-weight ratio of a drone related to its performance?
4. What assumptions are made in calculating the power scaling, and how might real-world factors alter this?
5. How does the efficiency of the drone's motors impact the power scaling with size?

Tip: When scaling objects in physics, remember that not all properties scale linearly—volume, surface area, and mass scale differently depending on the dimensions.