To determine the correct statement, we can use the relationship between force, mass, acceleration, and power.
According to Newton's second law, the force \( F \) applied to an object is given by:
\[ F = m \cdot a \]
Where:
- \( m \) is the mass of the object
- \( a \) is the acceleration of the object
Since both rockets have the same mass, let’s denote the mass of the rockets as \( m \).
If rocket A accelerates twice as quickly as rocket B, we can denote the acceleration of rocket B as \( a \) and the acceleration of rocket A as \( 2a \).
Now, using Newton's second law:
- The force exerted by rocket A (\( F_A \)) is:
\[ F_A = m \cdot (2a) = 2ma \]
- The force exerted by rocket B (\( F_B \)) is:
\[ F_B = m \cdot a \]
The power \( P \) produced by an engine is given by:
\[ P = F \cdot v \]
Where \( v \) is the velocity of the rocket. Since the acceleration of rocket A is double that of rocket B, it will also reach a greater speed more quickly, leading to a higher power requirement over the same timeframe if we consider them over similar distances, for instance.
However, for simplicity, if we assume both rockets reach the same speed at a certain point in time, we recognize that the work done (which is related to the power) would have to be greater for rocket A because it’s experiencing a greater force over the same distance in the same time.
Given that the force for rocket A is twice that of rocket B (\( F_A = 2 F_B \)), for the same time and distance, rocket A would need four times the power of rocket B when accounting for the relationship between force and power through velocity.
Therefore, the correct statement is:
The motor in rocket A is four times as powerful as the motor in rocket B.
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