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Chapter 9: Problem 25

A 92-kg astronaut and a 1200-kg satellite are at rest relative to the space shuttle. The astronaut pushes on the satellite, giving it a speed of \(0.14 \mathrm{m} / \mathrm{s}\) directly away from the shuttle. Seven and a half seconds later the astronaut comes into contact with the shuttle. What was the initial distance from the shuttle to the astronaut?

Short Answer

Expert verified
The initial distance from the shuttle to the astronaut was approximately 13.70 meters.

Step by step solution

01

Understand the Conservation of Momentum

The principle of conservation of momentum tells us that the total momentum of a system remains constant if no external forces are acting on it. In this scenario, because the astronaut and the satellite are initially at rest, their total initial momentum is zero. After the astronaut pushes the satellite, the momentum must still total zero, so the momentum gained by the satellite must be equal and opposite to the momentum gained by the astronaut.
02

Calculate Astronaut's Recoil Velocity

Since momentum is conserved, the momentum of the astronaut (mass = 92 kg) moving towards the shuttle should equal the momentum of the satellite (mass = 1200 kg) moving away. \[92 \cdot v = 1200 \cdot 0.14\] Solve for \(v\), the velocity of the astronaut: \[v = \frac{1200 \cdot 0.14}{92} = 1.826087 \, \mathrm{m/s}\]
03

Calculate the Time to Reach the Shuttle

The velocity we just calculated is also the velocity at which the astronaut is moving towards the shuttle. Given that the time taken is 7.5 seconds, use the formula for distance: distance = velocity 脳 time to calculate the distance covered by the astronaut:
04

Calculate Initial Distance from Shuttle to Astronaut

Now plug in the values into the distance formula: \[\text{distance} = 1.826087 \, \text{m/s} \times 7.5 \, \text{s} = 13.69565 \, \text{m}\] Thus, the astronaut was initially 13.69565 meters away from the shuttle.

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Key Concepts

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

Astronaut
The astronaut in our scenario plays a crucial role in demonstrating the principle of conservation of momentum. In space, astronauts are frequently involved in interactions with other objects, and their actions result in shifts in momentum due to the absence of frictional forces.
  • In our exercise, the astronaut weighs 92 kg and is initially at rest along with the satellite.
  • When the astronaut exerts a force on the satellite, it causes both the astronaut and the satellite to move in opposite directions, showcasing the concept of action and reaction forces as described by Newton's Third Law.
Understanding how astronauts interact with other objects is essential in space where maneuvering relies heavily on the principles of physics.
Satellite
Satellites are often objects of fascination when studying physics due to their unique behavior in the vacuum of space. In this exercise, the satellite weighs 1200 kg and is initially stationary with the astronaut.
  • Upon being pushed by the astronaut, the satellite receives a velocity of 0.14 m/s.
  • The mass and velocity relationship demonstrates how the satellite's movement obeys the momentum conservation in a closed system.
  • The satellite's mass is significantly larger than that of the astronaut, which means it gains less velocity for the same amount of momentum transfer.
Grasping satellite dynamics is vital, especially when considering orbital mechanics and space missions.
Recoil Velocity
Understanding recoil velocity is key in momentum conservation scenarios. After pushing the satellite, the astronaut experiences a recoil velocity due to the conservation principle.
  • The equation for calculating recoil velocity involves setting the momentum of the astronaut equal to that of the satellite.
  • By solving the equation \[92 \cdot v = 1200 \cdot 0.14\] we derive that the astronaut's recoil velocity is 1.826087 m/s.
  • This inverse relationship between mass and velocity illustrates how smaller objects gain greater velocity when momentum is transferred.
Recoil velocity is a splendid demonstration of how interactions in space can alter the motion of objects.
Distance Calculation
Calculating distance in physics often involves utilizing fundamental kinematic equations. For the astronaut to reach the shuttle, understanding their movement over time is necessary.
  • The astronaut's velocity (1.826087 m/s) and the time it takes (7.5 s) are crucial.
  • The formula to calculate distance is simply: \[\text{distance} = \text{velocity} \times \text{time}\]
  • By substituting the known values, the result is \[1.826087 \, \text{m/s} \times 7.5 \, \text{s} = 13.69565 \, \text{m}\].
This distance calculation succinctly ties together the astronaut's motion with the principles of momentum and velocity.

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Most popular questions from this chapter

A pencil standing upright on its eraser end falls over and lands on a table. As the pencil falls, its eraser does not slip. The following questions refer to the contact force exerted on the pencil by the table. Let the positive \(x\) direction be in the direction the pencil falls, and the positive \(y\) direction be vertically upward. (a) During the pencil's fall, is the \(x\) component of the contact force positive, negative, or zero? Explain. (b) Is the \(y\) component of the contact force greater than, less than, or equal to the weight of the pencil? Explain.

A 0.150-kg baseball is dropped from rest. If the magnitude of the baseball's momentum is \(0.780 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) just before it lands on the ground, from what height was it dropped?

Object A has a mass \(m\), object B has a mass \(2 m,\) and object C has a mass \(m / 2\). Rank these objects in order of increasing kinetic energy, given that they all have the same momentum. Indicate ties where appropriate.

Two air-track carts move toward one another on an air track. Cart 1 has a mass of \(0.35 \mathrm{kg}\) and a speed of \(1.2 \mathrm{m} / \mathrm{s}\). Cart 2 has a mass of \(0.61 \mathrm{kg}\). (a) What speed must cart 2 have if the total momentum of the system is to be zero? (b) Since the momentum of the system is zero, does it follow that the kinetic energy of the system is also zero? (c) Verify your answer to part (b) by calculating the system's kinetic energy.

Two ice skaters stand at rest in the center of an ice rink. When they push off against one another the \(45-\mathrm{kg}\) skater acquires a speed of \(0.62 \mathrm{m} / \mathrm{s}\). If the speed of the other skater is \(0.89 \mathrm{m} / \mathrm{s}\) what is this skater's mass?
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