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Chapter 3: Problem 44

If a \(100-\mathrm{kg}\) astronaut pushes on a \(5,000-\mathrm{kg}\) satellite and the satellite experiences an acceleration of \(0.1 \mathrm{m} / \mathrm{s}^{2},\) what is the acceleration experienced by the astronaut in the opposite direction?

Short Answer

Expert verified
The astronaut experiences an acceleration of 5 m/s^2 in the opposite direction.

Step by step solution

01

Understand Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. When the astronaut pushes on the satellite, the satellite pushes back on the astronaut with an equal and opposite force.
02

Calculate the Force on the Satellite

Use Newton's Second Law to find the force on the satellite. The formula is: \( F = m \times a \) where \( m \) is the mass of the satellite (5000 kg) and \( a \) is the acceleration (0.1 m/s^2). \( F = 5000 \times 0.1 = 500 \, \text{N} \)
03

Apply Newton's Third Law to the Astronaut

The force exerted on the astronaut is equal in magnitude and opposite in direction to the force on the satellite. Therefore, the force on the astronaut is 500 N, but in the opposite direction.
04

Calculate the Acceleration of the Astronaut

Using Newton's Second Law again, the acceleration of the astronaut can be found using the formula: \( a = \frac{F}{m} \) where \( F \) is the force on the astronaut (500 N) and \( m \) is the mass of the astronaut (100 kg). \( a = \frac{500}{100} = 5 \, \text{m/s}^2 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton's Third Law
Newton's Third Law is fundamental in understanding how forces work in pairs. This law states that for every action, there's an equal and opposite reaction. Imagine you're standing on a skateboard and you push against a wall. You move backward because the wall pushes back on you with an equal force. This principle is key in our exercise. When the astronaut pushes on the satellite, the satellite pushes back on the astronaut with the same force, just in the opposite direction. This explains why the astronaut feels a reaction force.
Newton's Second Law
Newton's Second Law tells us that the force acting on an object is equal to the mass of that object multiplied by its acceleration. The formula is given by \( F = m \times a \). To find the force on the satellite, we use this formula. With the given values, where the mass of the satellite is 5000 kg and the acceleration is 0.1 \(\text{m/s}^2\), we calculate: \( F = 5000 \times 0.1 = 500 \text{ N} \) This force is also the reaction force that the astronaut feels, as per Newton’s Third Law.
Force and Acceleration
In physics, understanding the relationship between force and acceleration is crucial. According to Newton's Second Law, the acceleration experienced by an object is the result of the force applied to it divided by its mass: \( a = \frac{F}{m} \). For the astronaut experiencing a force of 500 N with a mass of 100 kg, the acceleration can be calculated as: \( a = \frac{500}{100} = 5 \text{ m/s}^2 \). Thus, the astronaut will accelerate in the opposite direction to the push on the satellite with a magnitude of 5 \(\text{m/s}^2\). Knowing these concepts helps us understand how forces and movements are interconnected.

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