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Chapter 13: Problem 92

A \(150.0 \mathrm{~kg}\) rocket moving radially outward from Earth has a speed of \(3.70 \mathrm{~km} / \mathrm{s}\) when its engine shuts off \(200 \mathrm{~km}\) above Earth's surface. (a) Assuming negligible air drag acts on the rocket, find the rocket's kinetic energy when the rocket is \(1000 \mathrm{~km}\) above Earth's surface. (b) What maximum height above the surface is reached by the rocket?

Short Answer

Expert verified
The kinetic energy at 1000 km is found using conservation of energy, and the maximum height is found by setting kinetic energy to zero.

Step by step solution

01

Understand the Problem

We are given a rocket with a mass of 150 kg, initial velocity of 3.70 km/s, and initial altitude of 200 km. We need to find its kinetic energy at 1000 km altitude and maximum height considering negligible air drag.
02

Calculate Initial Kinetic Energy

Initial velocity \( v_0 = 3.70 \text{ km/s} = 3700 \text{ m/s} \). Using the formula for kinetic energy \( KE = \frac{1}{2} mv^2 \), where \( m = 150 \text{ kg} \), the initial kinetic energy is \( KE_0 = \frac{1}{2} \times 150 \times (3700)^2 \).
03

Setup the Conservation of Energy Equation

At any point, the total energy \(E\) is conserved: \(E = KE + U\), where \(U\) is gravitational potential energy given by \(U = -\frac{GMm}{r}\). \(r_0 = R_e + 200,000 \text{ m}\) and \(r = R_e + 1,000,000 \text{ m}\), where \(R_e\) is Earth's radius (6,371 km or 6,371,000 m).
04

Determine the Kinetic Energy at 1000 km

Using the conservation of energy: \[ \frac{1}{2}mv_0^2 - \frac{GMm}{r_0} = \frac{1}{2}mv^2 - \frac{GMm}{r} \], solve for \(v^2\) at \(r = R_e + 1,000,000 m\) and substitute to find the kinetic energy at this point.
05

Use the Conservation of Energy to Find the Maximum Height

At the maximum height, the kinetic energy is zero (\(v=0\)). Use \(\frac{1}{2}mv_0^2 - \frac{GMm}{r_0} = -\frac{GMm}{r_{max}}\) to solve for \(r_{max}\). The height above the surface is \(r_{max} - R_e\).

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Key Concepts

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

Kinetic Energy
Kinetic energy is a fundamental concept in physics that deals with the energy an object possesses due to its motion. When a rocket is moving through space, it carries kinetic energy dependent on its mass and speed.
The formula for kinetic energy (KE) is given by:
  • \( KE = \frac{1}{2} mv^2 \)
Here, \(m\) is the mass of the rocket, and \(v\) is its speed. To calculate the kinetic energy, simply plug the values of mass and velocity into the formula above. For instance, the given rocket of mass 150 kg, initially travelling at 3,700 meters per second, carries significant kinetic energy.
Through the journey, as the rocket ascends to 1,000 km above Earth's surface, this kinetic energy transforms as the velocity changes due to decreasing influence of Earth's gravity and increasing gravitational potential energy.
Gravitational Potential Energy
Gravitational potential energy (GPE) is the energy stored by objects due to their position relative to Earth. It is influenced by the mass of the object, the gravitational constant, and the distance from the center of Earth.
The formula for gravitational potential energy is:
  • \( U = -\frac{GMm}{r} \)
In this formula, \(G\) is the universal gravitational constant, \(M\) is Earth's mass, \(m\) is the mass of the rocket, and \(r\) is the distance from Earth's center. The negative sign indicates that gravity pulls the object towards Earth.
As the rocket climbs to higher altitudes, such as 1,000 km above Earth, the gravitational pull weakens as the distance \(r\) increases. This causes the GPE to increase, contrasting with the decrease in kinetic energy as the rocket slows down due to gravity's influence. At the rocket's maximum height, the GPE is at its peak, while kinetic energy reduces to zero.
Conservation of Energy
The principle of conservation of energy states that the total energy in a closed system remains constant over time. For the rocket in space, this means that its total energy, which is a combination of kinetic energy (KE) and gravitational potential energy (GPE), does not change, even as it moves to higher altitudes.
Mathematically, this can be expressed as:
  • \( \text{Total Energy} = KE + U = \text{constant} \)
In the given rocket problem, the initial total energy at 200 km above Earth's surface is comprised of KE at its initial speed and GPE due to its height.
As the rocket ascends further, up to 1,000 km, the energy manifests differently: the kinetic energy may lessen while the gravitational potential energy increases. At the maximum height, where the rocket momentarily stops ascending, all the initial kinetic energy has transformed into gravitational potential energy. This balance and transformation illustrate the conservation of energy principle perfectly.

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