To determine which statement is correct, we need to understand the relationship between power, force, and acceleration. According to Newton's second law, the force \( F \) acting on an object is equal to the mass \( m \) of the object times its acceleration \( a \):
\[ F = m \cdot a \]
Power \( P \) is defined as the rate of doing work or the rate at which energy is transferred or converted. It can also be expressed in terms of force and velocity:
\[ P = F \cdot v \]
Where:
- \( v \) is the velocity of the rocket.
Given that both rockets have the same mass and Rocket A accelerates twice as quickly as Rocket B, we can deduce the following:
- Let the mass of each rocket be \( m \).
- Let the acceleration of Rocket B be \( a \) (then the acceleration of Rocket A is \( 2a \)).
- According to Newton's second law, the force required for Rocket B is \( F_B = m \cdot a \) and the force required for Rocket A is \( F_A = m \cdot (2a) = 2m \cdot a \).
Rocket A's force is double that of Rocket B.
- Power is also related to acceleration and velocity. If we assume both rockets start from rest and after some time t they have the same time to reach their respective velocities, Rocket A, having higher acceleration, would achieve a higher final velocity faster than Rocket B.
Assuming both rockets experience constant acceleration, after a certain time \( t \):
- The velocity of Rocket A will be \( v_A = 2a \cdot t \)
- The velocity of Rocket B will be \( v_B = a \cdot t \)
Now substituting back into the power equations for both rockets:
- For Rocket A: \( P_A = F_A \cdot v_A = 2(m \cdot a) \cdot (2a \cdot t) = 4m \cdot a^2 \cdot t \)
- For Rocket B: \( P_B = F_B \cdot v_B = (m \cdot a) \cdot (a \cdot t) = m \cdot a^2 \cdot t \)
Now, comparing the two powers: \[ P_A = 4 P_B \]
Therefore, the correct statement is:
The motor in rocket A is four times as powerful as the motor in rocket B.
3 answers