A 85-kg astronaut pushes to the left on a spacecraft with a force F in “gravity-free” space. The spacecraft has a total mass of 1.0 × 10^4 kg. During the push, the astronaut accelerates to the right with an acceleration of 0.36 m/s^2.
Which one of the following statements concerning this situation is true?
A) The spacecraft does not move, but the astronaut moves to the right with a constant speed.
B) The astronaut stops moving after he stops pushing on the spacecraft.
C) The force exerted on the astronaut is larger than the force exerted on the spacecraft.
D) The force exerted on the spacecraft is larger than the force exerted on the astronaut.
E) The velocity of the astronaut increases while he is pushing on the spacecraft.
Determine the magnitude of the acceleration of the spacecraft.
A) 30.6 m/s2
B) 0.36 m/s2
C) 2.5 × 10–3 m/s2
D) 7.0 × 10–3 m/s2
E) 3.06 × 10–3 m/s2
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To address the first part of the question, let's analyze each statement:
A) Incorrect. According to Newton's third law, for every action, there is an equal and opposite reaction. If the astronaut pushes on the spacecraft, the spacecraft will move in the opposite direction (to the left, in this case), not remain stationary.
B) Incorrect. Even after the astronaut stops pushing, both the astronaut and the spacecraft will continue moving at constant velocities due to the conservation of momentum, as there are no external forces acting on the system (assuming "gravity-free" space implies no external forces).
C) Incorrect. According to Newton's third law, the force exerted by the astronaut on the spacecraft is equal in magnitude and opposite in direction to the force exerted by the spacecraft on the astronaut.
D) Incorrect for the same reason as C. The forces are equal in magnitude.
E) Correct. While the astronaut is pushing on the spacecraft and experiencing an acceleration, his velocity increases over time according to the equation \(v = at\), where \(v\) is velocity, \(a\) is acceleration, and \(t\) is time.
For the second part, to find the acceleration of the spacecraft, we use Newton's second law, \(F = ma\), where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration. Since the force exerted by the astronaut on the spacecraft is equal in magnitude to the force exerted by the spacecraft on the astronaut, we can find the force exerted by using the astronaut's mass and acceleration.
Given:
- Mass of astronaut (\(m_a\)) = 85 kg
- Acceleration of astronaut (\(a_a\)) = 0.36 m/s^2
The force exerted by the astronaut (and thus on the spacecraft) is:
\[F = m_a \times a_a = 85 \, \text{kg} \times 0.36 \, \text{m/s}^2 = 30.6 \, \text{N}\]
To find the acceleration of the spacecraft (\(a_s\)), we rearrange the formula to \(a_s = \frac{F}{m_s}\), where \(m_s\) is the mass of the spacecraft.
Given:
- Mass of spacecraft (\(m_s\)) = 1.0 × 10^4 kg
\[a_s = \frac{30.6 \, \text{N}}{1.0 \times 10^4 \, \text{kg}} = 3.06 \times 10^{-3} \, \text{m/s}^2\]
Therefore, the correct answers are:
- E) The velocity of the astronaut increases while he is pushing on the spacecraft.
- E) 3.06 × 10^–3 m/s^2 for the magnitude of the acceleration of the spacecraft.
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