Question

An 80 -kg astronaut becomes separated from his spaceship. He is \(15.0 \mathrm{~m}\) away from it and at rest relative to it. In an effort to get back, he throws a \(500-\mathrm{g}\) object with a speed of \(8.0 \mathrm{~m} / \mathrm{s}\) in a direction away from the ship. How long does it take him to get back to the ship? a) \(1 \mathrm{~s}\) b) \(10 \mathrm{~s}\) c) \(20 \mathrm{~s}\) d) \(200 \mathrm{~s}\) e) \(300 \mathrm{~s}\)

Short answer

Answer: (e) 300 s

Step-by-step solution

Calculate the astronaut's initial momentum

Before the astronaut throws the object, their initial momentum is zero since they are at rest. The object also starts from rest, so its initial momentum is also zero.

Calculate the object's final momentum

After the astronaut throws the object, we can calculate its final momentum using the formula: momentum = mass x velocity For the object, we have: mass = 0.5 kg (we convert 500 g to kilograms), and velocity = 8.0 m/s: momentum\_object = 0.5 kg × 8.0 m/s = 4.0 kg m/s

Calculate the astronaut's final momentum

Since the total momentum is conserved, the final momentum of the astronaut must be equal and opposite to the final momentum of the object: momentum\_astronaut = -momentum\_object = -4.0 kg m/s

Calculate the astronaut's final velocity

Now we will determine the astronaut's final velocity using the momentum formula: velocity\_astronaut = momentum\_astronaut / mass\_astronaut where mass\_astronaut = 80 kg: velocity\_astronaut = -4.0 kg m/s / 80 kg = -0.05 m/s Note that the velocity is negative because the astronaut is moving in the opposite direction of the thrown object.

Calculate the time it takes for the astronaut to reach the spaceship

To find the time it takes for the astronaut to cover the 15.0 m distance back to the spaceship, we will use the kinematic equation: distance = velocity x time Solving for time, we get: time = distance / velocity Using the values for the distance (15.0 m) and the astronaut's final velocity (-0.05 m/s), we find the time: time = 15.0 m / -0.05 m/s = 300 s So, it takes the astronaut 300 seconds to reach the spaceship. The correct answer is (e) \(300 \mathrm{~s}\).

Key concepts

Momentum

Momentum is a fundamental concept in physics, representing the quantity of motion an object possesses. It is a vector quantity, meaning it has both magnitude and direction, and is calculated by the product of an object's mass and velocity, represented by the equation: \( \text{momentum} = \text{mass} \times \text{velocity} \). In any closed system, the total momentum before an event is equal to the total momentum after the event, given there are no external forces. This principle is known as the conservation of momentum.

In the problem presented, the astronaut and the object together form a closed system. Initially, both are at rest, and thus the total initial momentum of the system is zero. After the astronaut throws the object, the system's total momentum must still be zero. Since the object gains momentum in one direction, the astronaut will gain an equal amount of momentum in the opposite direction to maintain the conservation of momentum.

Kinematic Equations

Kinematic equations describe the motion of objects without considering the forces that cause the motion. They are crucial in solving physics problems involving velocity, acceleration, time, and displacement. One common kinematic equation is \( \text{distance} = \text{velocity} \times \text{time} \), which shows the direct relationship between distance traveled, the constant velocity of an object, and the time taken.

In the astronaut's scenario, we use this kinematic equation to find how long it will take him to cover the distance to the spaceship. Since his velocity is constant after throwing the object, and we know the distance to be covered, the equation simplifies to determining the time it takes him to reach the spaceship. It is essential to include the proper signs for velocity to indicate the direction of movement, as negative velocity indicates motion in the direction opposite to the positive reference direction.

Physics Problem Solving

Physics problem solving often involves breaking down complex situations into more manageable parts and providing solutions that utilize fundamental concepts like conservation of momentum and kinematic equations. A systematic approach, as seen in this astronaut example, follows these steps: identify the relevant concepts, extract and convert given information into the proper units, apply equations logically, and solve for the unknowns.

It's important to recognize that in physics, both magnitude and direction matter, so attention to detail when working with vector quantities is crucial. In our example, the astronaut's action of throwing the object results in his movement in the opposite direction, which is why the concept of direction is essential in determining the solution. Keeping a clear and organized approach, as demonstrated in this problem, aids in reducing errors and developing a deeper understanding of the subject.

Astronaut Physics

Astronaut physics offers unique challenges due to the microgravity environment of space. Because there is no gravity to influence motion, principles like conservation of momentum become very visible and crucial for maneuvers. Astronauts must use the forces generated by their own actions or reactions from their equipment to move around, as demonstrated by the astronaut throwing an object to propel himself back to his spaceship.

The absence of external forces, such as friction, makes this a perfect example of Newton's third law, which states that for every action, there is an equal and opposite reaction. As a result, when the astronaut throws the object in one direction, he moves in the opposite direction. This mode of transportation, referred to as propulsive movement, is a fundamental skill for astronauts maneuvering in space.