To determine the power of the motors in rockets A and B, we can use the relationship between force, acceleration, and power.
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From Newton's second law, we know that \( F = ma \), where \( F \) is the force applied, \( m \) is the mass of the rocket, and \( a \) is the acceleration.
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If rocket A accelerates twice as quickly as rocket B, we can denote the acceleration of rocket B as \( a \) and that of rocket A as \( 2a \).
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Since both rockets have the same mass \( m \):
- For rocket A: \( F_A = m(2a) = 2ma \)
- For rocket B: \( F_B = ma \)
Therefore, the force in rocket A is twice the force in rocket B: \( F_A = 2F_B \).
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Power is defined as the rate at which work is done or the rate at which energy is transferred. Power can be calculated using the formula \( P = Fv \), where \( v \) is the velocity.
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For the situation where both rockets are accelerating, if we say at some point in time rocket A reaches a velocity \( v_A \) and rocket B reaches a velocity \( v_B \), the power of each rocket can be expressed as:
- \( P_A = F_A v_A \)
- \( P_B = F_B v_B \)
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Because rocket A accelerates twice as fast, we might expect that it could also reach a proportional higher velocity, depending on the time interval in consideration. However, without loss of generality, if we assume they are both at the same instant under equal conditions for comparison (which is a common assumption in physics problems for simplification), the ratio of their powers will depend on the forces exerted due to the accelerations.
Since \( F_A = 2F_B \) and if we assumed that both reach the same velocity at the moment of comparison, then:
- \( P_A = 2F_B v \)
- \( P_B = F_B v \)
- Hence, \( P_A = 2P_B \).
Thus, the correct statement is:
The motor in rocket A is twice as powerful as the motor in rocket B.