To determine the correct statement regarding the power of the motors in rockets A and B, we can use the relationship between force, acceleration, mass, and power.
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Let's denote:
- Mass of Rocket A (m_A) = Mass of Rocket B (m_B) = m (since they have the same mass)
- Acceleration of Rocket A (a_A) = 2a (twice as quick as B)
- Acceleration of Rocket B (a_B) = a
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According to Newton's second law, the force exerted by each rocket can be expressed as:
- \( F_A = m \cdot a_A = m \cdot (2a) = 2ma \) (for Rocket A)
- \( F_B = m \cdot a_B = m \cdot a \) (for Rocket B)
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The power (P) used by a force can be described as: \[ P = F \cdot v \] where v is the velocity.
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The relationship between force and acceleration gives:
- To achieve acceleration, the force generated by the rocket must be proportional to that acceleration.
In this case, if Rocket A has twice the acceleration and the same mass, it must generate twice the force compared to Rocket B. Since the power output is related to the force and the velocity of the objects, and since both rockets are assumed to be under similar operating conditions (say starting from rest and moving to the same velocity), we can conclude that:
- If Rocket A accelerates twice as quickly as Rocket B, it means it must do twice the work in the same amount of time (assuming the same distance or velocity reached).
Thus, the correct statement is:
The motor in rocket A is twice as powerful as the motor in rocket B.
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