To determine the correct statement regarding the power of the motors in the two rockets, we need to use the relationship between force, acceleration, and power.
- Newton's Second Law: The force required to accelerate an object is given by \( F = ma \), where \( m \) is the mass and \( a \) is the acceleration.
- Since both rockets have the same mass, for rocket A, which accelerates twice as quickly as rocket B, we can express the forces as follows:
- Let \( F_A \) be the force exerted by rocket A and \( F_B \) be the force exerted by rocket B.
- If rocket B has an acceleration \( a \), then rocket A has an acceleration \( 2a \).
Thus, we have:
- \( F_A = m \cdot 2a = 2ma \)
- \( F_B = m \cdot a = ma \)
- Power is given by \( P = F \cdot v \), where \( v \) is the velocity. Assuming both rockets start from rest and rocket A reaches twice the acceleration, it will also have a higher velocity at any given time compared to rocket B.
However, for the sake of immediate comparison of their power while maintaining the same velocity relationship (assuming they reach similar velocities), we can simply consider the relationship between force and power since we're looking for a comparative statement without the need for exact velocities.
Since \( F_A = 2 F_B \) (because \( F_A = 2ma \) and \( F_B = ma \)):
- If \( P_A = F_A \cdot v_A \)
- And \( P_B = F_B \cdot v_B \)
Setting \( v_A = 2v_B \) (if we momentarily consider they reach double velocities corresponding to their accelerations at some point):
- Then \( P_A = 2F_B \cdot 2v_B = 4F_B \cdot v_B = 4P_B\)
Conclusion:
The motor in rocket A is four times as powerful as the motor in rocket B.
Correct Statement: The motor in rocket A is four times as powerful as the motor in rocket B.