Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

Problem 17

Chapter 13, Problem 17

A rocket is launched straight up from the earth's surface at a speed of 15,000 m/s. What is its speed when it is very far away from the earth?

Verified step by step guidance

Step 1: Recognize that this problem involves the conservation of energy principle. The rocket's initial kinetic energy and gravitational potential energy will be converted into its final kinetic energy when it is very far away from the Earth.

Step 2: Write the total mechanical energy equation. The initial total energy is the sum of the rocket's kinetic energy and gravitational potential energy: \( E_{initial} = \frac{1}{2}mv^2 - \frac{GMm}{R} \), where \( v \) is the initial velocity, \( G \) is the gravitational constant, \( M \) is the Earth's mass, \( R \) is the Earth's radius, and \( m \) is the rocket's mass.

Step 3: At a very far distance from the Earth, the gravitational potential energy approaches zero, and the rocket's final total energy is purely kinetic: \( E_{final} = \frac{1}{2}mv_{final}^2 \).

Step 4: Apply the conservation of energy principle: \( E_{initial} = E_{final} \). Substitute the expressions for \( E_{initial} \) and \( E_{final} \): \( \frac{1}{2}mv^2 - \frac{GMm}{R} = \frac{1}{2}mv_{final}^2 \).

Step 5: Simplify the equation to solve for \( v_{final} \): \( v_{final} = \sqrt{v^2 - \frac{2GM}{R}} \). Substitute the given values for \( v \), \( G \), \( M \), and \( R \) to calculate the final speed.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

11m

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Energy and Gravitational Potential Energy

In the context of a rocket launch, kinetic energy is the energy of motion, while gravitational potential energy is the energy stored due to an object's position in a gravitational field. As the rocket ascends, its kinetic energy decreases while its gravitational potential energy increases until it reaches a point where the gravitational influence of the Earth is negligible.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. For the rocket, the initial kinetic energy at launch will convert into gravitational potential energy as it rises, and when it is far from Earth, this energy will primarily be kinetic, allowing us to calculate its speed at that point.

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from a celestial body's gravitational pull without further propulsion. For Earth, this speed is approximately 11,186 m/s. Since the rocket's initial speed exceeds this threshold, it will continue to move away from Earth, and its speed will be determined by the conservation of energy as it moves into space.

Recommended video:

Guided course

7:27

Escape Velocity

Related Practice

Textbook Question

You have been visiting a distant planet. Your measurements have determined that the planet's mass is twice that of earth but the free-fall acceleration at the surface is only one-fourth as large. What is the planet's radius?

views

Textbook Question

A space station orbits the sun at the same distance as the earth but on the opposite side of the sun. A small probe is fired away from the station. What minimum speed does the probe need to escape the solar system?

views

Textbook Question

Suppose we could shrink the earth without changing its mass. At what fraction of its current radius would the free-fall acceleration at the surface be three times its present value?

views

Textbook Question

Two stars, one twice as massive as the other, are 1.0 light year (ly) apart. One light year is the distance light travels in one year at the speed of light, 3.00 ✕ 10⁸ m/s . The gravitational potential energy of this double-star system is - 8.0 ✕ 10³⁴ J. What is the mass of the lighter star?

Textbook Question

Asteroid 253 Mathilde is one of several that have been visited by space probes. This asteroid is roughly spherical with a diameter of 53 km. The free-fall acceleration at the surface is 9.9 ✕ 10^-3 m/s². What is the asteroid's mass?

views

Textbook Question

What is the escape speed from Jupiter?