To determine the correct statement, we can use Newton's second law of motion, which states that \( F = ma \) (force equals mass times acceleration). If both rockets have the same mass and Rocket A accelerates twice as quickly as Rocket B, we can analyze the forces and power involved.
Let:
- \( m \) = mass of each rocket
- \( a_A \) = acceleration of Rocket A
- \( a_B \) = acceleration of Rocket B
Given:
- \( a_A = 2a_B \)
The force exerted by each rocket can be expressed as:
- For Rocket A: \( F_A = ma_A = m(2a_B) = 2ma_B \)
- For Rocket B: \( F_B = ma_B \)
To compare the powers of the motors, recall that power is the rate of doing work or the force times the velocity. However, since both rockets have the same mass and Rocket A accelerates twice as quickly, the motor's power can also be related directly to the force.
The force needed to achieve these accelerations shows that:
- \( F_A = 2F_B \)
Thus, we see that the motor in Rocket A must generate twice the force of the motor in Rocket B.
Now, regarding power: Power can also be described as \( P = Fv \) where \( v \) is the velocity. Since Rocket A accelerates faster, and assuming both rockets operate under similar conditions leading to equivalent velocities proportional to their accelerations, Rocket A would require more power to maintain its acceleration.
Thus since Rocket A exerts twice the force, it also implies that its motor is twice as powerful.
The correct statement is: The motor in rocket A is twice as powerful as the motor in rocket B.