To determine the correct statement regarding the power of the motors in rockets A and B, we need to understand the relationship between force, acceleration, and power.
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Newton's Second Law of Motion states that \( F = ma \), where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration. Since both rockets have the same mass, the force required to accelerate each rocket is directly proportional to the acceleration.
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Since Rocket A accelerates twice as quickly as Rocket B, we can say:
- If the acceleration of Rocket B is \( a \), then the acceleration of Rocket A is \( 2a \).
- The force for Rocket A is therefore \( F_A = m \cdot (2a) = 2ma \).
- The force for Rocket B is \( F_B = ma \).
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The power (\( P \)) of the motors is given by the formula \( P = Fv \), where \( v \) is the velocity. In this case, we can focus on the fact that power is also proportional to force when we consider the instantaneous power just as the rockets are initiated.
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Since Rocket A produces a force that is twice that of Rocket B, we can conclude that the power generated by Rocket A's motor is also greater.
When Rocket A is accelerating at twice the rate, it means:
- \( P_A = F_A v_A \)
- \( P_B = F_B v_B \)
As both rockets have the same mass and are accelerating, we find the relationship of power by comparing the forces:
- Since \( P_A \propto F_A\) and \( P_B \propto F_B \)
- \( F_A = 2F_B \), we get \( P_A = 2P_B \) when we talk about average powers for equal time intervals.
Therefore, the correct statement is:
The motor in rocket A is twice as powerful as the motor in rocket B.
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