Skip to main content
Ch 13: Gravitation
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 13: GravitationProblem 19
Chapter 13, Problem 19

A planet orbiting a distant star has radius 3.24 × 106 m. The escape speed for an object launched from this planet’s surface is 7.65 × 103 m/s. What is the acceleration due to gravity at the surface of the planet?

Verified step by step guidance
1
Start by recalling the formula for escape speed, which is given by \( v_e = \sqrt{2gr} \), where \( v_e \) is the escape speed, \( g \) is the acceleration due to gravity, and \( r \) is the radius of the planet.
Rearrange the formula to solve for \( g \): \( g = \frac{v_e^2}{2r} \).
Substitute the given values into the equation: \( v_e = 7.65 \times 10^3 \text{ m/s} \) and \( r = 3.24 \times 10^6 \text{ m} \).
Calculate \( v_e^2 \) by squaring the escape speed: \( (7.65 \times 10^3)^2 \).
Divide the result of \( v_e^2 \) by \( 2r \) to find the acceleration due to gravity \( g \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Escape Velocity

Escape velocity is the minimum speed an object must reach to break free from a celestial body's gravitational pull without further propulsion. It depends on the mass and radius of the body. For a planet, the escape velocity (v) is given by the formula v = sqrt(2 * G * M / R), where G is the gravitational constant, M is the planet's mass, and R is its radius.
Recommended video:
Guided course
7:27
Escape Velocity

Gravitational Acceleration

Gravitational acceleration (g) is the acceleration experienced by an object due to the gravitational force exerted by a massive body like a planet. It is calculated using the formula g = G * M / R^2, where G is the gravitational constant, M is the mass of the planet, and R is its radius. This acceleration determines the weight of objects on the planet's surface.
Recommended video:
Guided course
07:32
Weight Force & Gravitational Acceleration

Relationship Between Escape Velocity and Gravitational Acceleration

The escape velocity and gravitational acceleration are related through the planet's mass and radius. By equating the escape velocity formula v = sqrt(2 * G * M / R) with the gravitational acceleration formula g = G * M / R^2, we can derive g = v^2 / (2 * R). This relationship allows us to calculate the gravitational acceleration if the escape velocity and radius are known.
Recommended video:
Guided course
7:27
Escape Velocity
Related Practice
Textbook Question

Jupiter's moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.93 × 1022 kg and a radius of 1821 km. For this calculation, ignore any variation in gravity over the 500-km range of the debris. How high would this material go on earth if it were ejected with the same speed as on Io?

5078
views
1
comments
Textbook Question

Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was 2.87 × 106 km from the earth and traveling at 1.20 × 104 km/h relative to the earth. At this time, what were (a) the spacecraft's kinetic energy relative to the earth and (b) the potential energy of the earth–spacecraft system?

2878
views
Textbook Question

Use the results of Example 13.5 (Section 13.3) to calculate the escape speed for a spacecraft (a) from the surface of Mars and (b) from the surface of Jupiter. Use the data in Appendix F. (c) Why is the escape speed for a spacecraft independent of the spacecraft's mass?

1527
views
Textbook Question

Two satellites are in circular orbits around a planet that has radius 9.00 × 106 m. One satellite has mass 68.0 kg, orbital radius 7.00 × 107 m, and orbital speed 4800 m/s. The second satellite has mass 84.0 kg and orbital radius 3.00 × 107 m. What is the orbital speed of this second satellite?

3195
views
Textbook Question

For a satellite to be in a circular orbit 890 km above the surface of the earth, what orbital speed must it be given?

1967
views
Textbook Question

On July 15, 2004, NASA launched the Aura spacecraft to study the earth's climate and atmosphere. This satellite was injected into an orbit 705 km above the earth's surface. Assume a circular orbit. How many hours does it take this satellite to make one orbit?

2708
views