To determine the power of the motors in the two rockets based on their acceleration, we can use Newton's second law and the definition of power.
- According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration: \( F = m \cdot a \).
- The power (\( P \)) required to do work is given by the formula: \( P = F \cdot v \), where \( v \) is velocity.
Given that both rockets have the same mass \( m \), let's denote the acceleration of Rocket B as \( a \). Therefore, the acceleration of Rocket A is \( 2a \) (twice as much).
Now, we can write the forces for each rocket:
- Force for Rocket A: \( F_A = m \cdot (2a) = 2ma \)
- Force for Rocket B: \( F_B = m \cdot a = ma \)
Since power is also the product of force and velocity, we need to consider that typically, in a steady-state condition (for example, when the rockets reach a certain point after some time), the velocity of the rockets will be proportional to acceleration and time. If we assume they have been accelerating for the same duration \( t \):
- Velocity of Rocket A after time \( t \): \( v_A = 2a \cdot t \)
- Velocity of Rocket B after time \( t \): \( v_B = a \cdot t \)
Now we can calculate the power:
- Power for Rocket A: \( P_A = F_A \cdot v_A = (2ma)(2at) = 4ma^2t \)
- Power for Rocket B: \( P_B = F_B \cdot v_B = (ma)(at) = ma^2t \)
Now, comparing the powers:
- \( P_A = 4P_B \)
So, the motor in Rocket A is four times as powerful as the motor in Rocket B.
The correct statement is: The motor in rocket A is four times as powerful as the motor in rocket B.
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