Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 39

The escape speed from the surface of Planet Zoroaster is $12.0 \mathrm{km} / \mathrm{s} .$ The planet has no atmosphere. A meteor far away from the planet moves at speed \(5.0 \mathrm{km} / \mathrm{s}\) on a collision course with Zoroaster. How fast is the meteor going when it hits the surface of the planet?

Short Answer

Expert verified
Answer: The final speed of the meteor when it hits the surface of Planet Zoroaster is 13.0 km/s.

Step by step solution

01

Identify the variables

We are given the escape speed from the surface of the planet (\(v_e = 12.0\,\mathrm{km/s}\)) and the initial speed of the meteor far away from the planet (\(v_i = 5.0\,\mathrm{km/s}\)). We need to find the final speed of the meteor \(v_f\) on the surface of the planet.
02

Express the energy conservation law

According to the conservation of mechanical energy, the sum of the kinetic energy and gravitational potential energy remains constant throughout the motion. Let \(K_i\) and \(U_i\) be the initial kinetic energy and gravitational potential energy, and \(K_f\) and \(U_f\) be the final kinetic energy and gravitational potential energy. Then, we can express the conservation of mechanical energy as: \(K_i + U_i = K_f + U_f\)
03

Write the equations for kinetic and gravitational potential energies

Now we can write the kinetic energies (\(K\)) and gravitational potential energies (\(U\)) as functions of the speeds (\(v\)) and distances (\(r\)) from the center of the planet. Use the mass of the meteor (\(m\)) and the mass of Planet Zoroaster (\(M\)). Note that escape velocity (\(v_e\)) is related to the planet's mass and radius (\(R\)) by \(v_e^2 = 2GM/R\): Initial kinetic energy: \(K_i = \frac{1}{2}m v_i^2\) Initial gravitational potential energy: \(U_i \approx 0\) (Since the meteor is far away from the planet) Final kinetic energy: \(K_f = \frac{1}{2}m v_f^2\) Final gravitational potential energy: \(U_f = -\frac{GMm}{R}\)
04

Substitute the energies into the conservation of mechanical energy equation

Now, we can substitute the expressions for kinetic and gravitational potential energies into the conservation equation: \(\frac{1}{2}m v_i^2 + 0 = \frac{1}{2}m v_f^2 - \frac{GMm}{R}\) Note that the mass of the meteor \(m\) can be canceled out from both sides of the equation: \(\frac{1}{2} v_i^2 = \frac{1}{2} v_f^2 - \frac{GM}{R}\)
05

Solve for final speed \(v_f\) using escape velocity relation

Recall the escape velocity relation \(v_e^2 = 2GM/R\), and use it to solve for final velocity \(v_f\): \(v_i^2 = v_f^2 - v_e^2\) Now solve for \(v_f\): \(v_f = \sqrt{v_i^2 + v_e^2}\)
06

Calculate the final speed of the meteor

Substitute the given initial speed \(v_i = 5.0\,\mathrm{km/s}\) and escape speed \(v_e = 12.0\,\mathrm{km/s}\) into the equation for the final speed \(v_f\): \(v_f = \sqrt{(5.0\,\mathrm{km/s})^2 + (12.0\,\mathrm{km/s})^2}\) \(v_f = \sqrt{169\,\mathrm{km^{2}/s^{2}}} = 13\,\mathrm{km/s}\) So, the final speed of the meteor when it hits the surface of Planet Zoroaster is \(13.0\,\mathrm{km/s}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Jane is running from the ivory hunters in the jungle. Cheetah throws a 7.0 -m-long vine toward her. Jane leaps onto the vine with a speed of $4.0 \mathrm{m} / \mathrm{s} .$ When she catches the vine, it makes an angle of \(20^{\circ}\) with respect to the vertical. (a) When Jane is at her lowest point, she has moved downward a distance \(h\) from the height where she originally caught the vine. Show that \(h\) is given by \(h=L-L \cos 20^{\circ},\) where \(L\) is the length of the vine. (b) How fast is Jane moving when she is at the lowest point in her swing? (c) How high can Jane swing above the lowest point in her swing?
In \(1899,\) Charles \(\mathrm{M}\). "Mile a Minute" Murphy set a record for speed on a bicycle by pedaling for a mile at an average of $62.3 \mathrm{mph}(27.8 \mathrm{m} / \mathrm{s})$ on a 3 -mi track of plywood planks set over railroad ties in the draft of a Long Island Railroad train. In \(1985,\) a record was set for this type of "motor pacing" by Olympic cyclist John Howard who raced at 152.2 mph \((68.04 \mathrm{m} / \mathrm{s})\) in the wake of a race car at Bonneville Salt Flats. The race car had a modified tail assembly designed to reduce the air drag on the cyclist. What was the kinetic energy of the bicycle plus rider in each of these feats? Assume that the mass of bicycle plus rider is \(70.5 \mathrm{kg}\) in each case.
A shooting star is a meteoroid that burns up when it reaches Earth's atmosphere. Many of these meteoroids are quite small. Calculate the kinetic energy of a meteoroid of mass \(5.0 \mathrm{g}\) moving at a speed of $48 \mathrm{km} / \mathrm{s}$ and compare it to the kinetic energy of a 1100 -kg car moving at \(29 \mathrm{m} / \mathrm{s}(65 \mathrm{mi} / \mathrm{h})\).
Jorge is going to bungee jump from a bridge that is \(55.0 \mathrm{m}\) over the river below. The bungee cord has an unstretched length of \(27.0 \mathrm{m} .\) To be safe, the bungee cord should stop Jorge's fall when he is at least $2.00 \mathrm{m}\( above the river. If Jorge has a mass of \)75.0 \mathrm{kg},$ what is the minimum spring constant of the bungee cord?
(a) How much work does a major-league pitcher do on the baseball when he throws a \(90.0 \mathrm{mi} / \mathrm{h}(40.2 \mathrm{m} / \mathrm{s})\) fastball? The mass of a baseball is 153 g. (b) How many fastballs would a pitcher have to throw to "burn off' a 1520 -Calorie meal? (1 Calorie \(=1000\) cal \(=1\) kcal.)Assume that \(80.0 \%\) of the chemical energy in the food is converted to thermal energy and only \(20.0 \%\) becomes the kinetic energy of the fastballs.
See all solutions

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Mechanics and Materials

Read Explanation

Physical Quantities And Units

Read Explanation

Electricity and Magnetism

Read Explanation

Space Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free