To determine the relationship between the power of the rockets and their acceleration, we can use the formula for power:
\[ \text{Power} = \text{Force} \times \text{Velocity} \]
We also know from Newton's second law that:
\[ \text{Force} = \text{Mass} \times \text{Acceleration} \]
Since both rockets have the same mass, let's denote their mass as \( m \).
If Rocket A accelerates twice as quickly as Rocket B, we can denote their accelerations as:
- \( a_A = 2a_B \)
Using Newton's second law, the force exerted by each rocket can be expressed as:
- For Rocket A: \( F_A = m \cdot a_A = m \cdot (2a_B) = 2ma_B \)
- For Rocket B: \( F_B = m \cdot a_B \)
Now the ratio of the forces is:
\[ \frac{F_A}{F_B} = \frac{2ma_B}{ma_B} = 2 \]
Since power is the product of force and velocity and, at least for a moment assumed constant, velocity can be treated equally for both rockets when comparing their initial accelerations. If we assume the greater acceleration of Rocket A allows it to gain a certain velocity over a fixed time frame compared to Rocket B, we can infer that the average power over that time period will be additionally driven by the increase in the force applied.
Thus, considering that Rocket B has to catch up and given the doubling of force from A to in context of velocity dynamics resulting from the time, the power needed for Rocket A—accounting the increased force—will actually be:
\[ P_A = F_A \cdot v_A \] \[ P_B = F_B \cdot v_B \]
With \( F_A \) being \( 2F_B \) and given that \( v_A \) at the initial push increases faster given it relates to the squared input of force over time, this can lead us to derive that the engines of Rocket A must be:
- 4 times the power of Rocket B's motor.
Thus, the correct statement is:
The motor in rocket A is four times as powerful as the motor in rocket B.