Skip to main content
Ch 13: Newton's Theory of Gravity
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 13: Newton's Theory of GravityProblema 45
Capitolo 13, Problema 45

Suppose that on earth you can jump straight up a distance of 45 cm. Asteroids are made of material with mass density 2800 kg/m³ . What is the maximum diameter of a spherical asteroid from which you could escape by jumping?

Guida verificata passo dopo passo
1
Step 1: Begin by understanding the concept of escape velocity. Escape velocity is the minimum speed needed for an object to escape the gravitational pull of a celestial body without further propulsion. For a spherical asteroid, the escape velocity is determined by its mass and radius.
Step 2: Write the formula for escape velocity: vescape = √(2GM/R), where G is the gravitational constant (6.674 × 10-11 N·m²/kg²), M is the mass of the asteroid, and R is its radius.
Step 3: Relate the mass of the asteroid to its density and volume. The mass M can be expressed as M = ρV, where ρ is the density of the asteroid (2800 kg/m³) and V is its volume. For a sphere, the volume is given by V = (4/3)πR³.
Step 4: Substitute M = ρ(4/3)πR³ into the escape velocity formula. This gives vescape = √(2Gρ(4/3)πR²). Simplify the expression to find the relationship between escape velocity and radius.
Step 5: Set the escape velocity equal to the maximum jump velocity on Earth. Use the kinematic equation v = √(2gh), where g is the acceleration due to gravity on Earth (9.8 m/s²) and h is the jump height (0.45 m). Solve for R to find the maximum diameter of the asteroid.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula GPE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. In the context of jumping, the maximum height reached is directly related to the GPE converted from the kinetic energy of the jump.
Video consigliato:
Percorso guidato
06:35
Gravitational Potential Energy

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body without further propulsion. It depends on the mass and radius of the body, given by the formula v = √(2GM/R), where G is the gravitational constant, M is the mass of the body, and R is its radius. Understanding escape velocity is crucial for determining how high one must jump to escape an asteroid's gravity.
Video consigliato:
Percorso guidato
7:27
Escape Velocity

Mass Density

Mass density is defined as mass per unit volume, typically expressed in kg/m³. It is a critical factor in determining the mass of an object when its volume is known. For the asteroid in question, knowing its mass density allows us to calculate its mass based on its volume, which is essential for applying the escape velocity formula and understanding the gravitational effects on jumping.
Video consigliato:
Percorso guidato
04:33
Problems with Mass, Volume, & Density
Pratica correlata
Domanda del libro di testo

Two spherical objects have a combined mass of 150 kg. The gravitational attraction between them is 8.00 x 10-6 N when their centers are 20 cm apart. What is the mass of each?

1967
views
Domanda del libro di testo

Two spherical objects have a combined mass of 400 kg. The gravitational attraction between them is 6.00 x 10-7 N and their gravitational potential energy is ₋1.20 x 10-6 J. What is the mass of each?

306
views
Domanda del libro di testo

A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet's surface. How far above the surface are they orbiting? Your answer will be a multiple of R.

1275
views
Domanda del libro di testo

A rogue band of colonists on the moon declares war and prepares to use a catapult to launch large boulders at the earth. Assume that the boulders are launched from the point on the moon nearest the earth. For this problem you can ignore the rotation of the two bodies and the orbiting of the moon. What is the minimum speed with which a boulder must be launched to reach the earth? Hint: The minimum speed is not the escape speed. You need to analyze a three-body system.

1298
views
Domanda del libro di testo

The two stars in a binary star system have masses 2.0 x 10³⁰ kg and 6.0 x 10³⁰ kg. They are separated by 2.0 x 10¹² m. What are The speed of each star?

388
views
Domanda del libro di testo

What is the total gravitational potential energy of the three masses in FIGURE P13.36?

1116
views