---# Escape Velocity and TrajectoryConsider the process of escaping from the solar system starting from the surface of Earth. Assume there are no other bodies involved. Earth has an orbital speed about the Sun of 29.8 km/s. Use the following equation which includes the potential energy of both Earth and the Sun:\[ v_{\text{esc}} = \sqrt{\frac{2GM}{R}} \]does not apply. Use:\[ \frac{1}{2} mv_1^2 - \frac{GM_E m}{r_{1,E}} = \frac{1}{2} mv_2^2 - \frac{GM_E m}{r_{2,E}} - \frac{GM_S m}{r_{2,S}} \]### (a) What minimum speed relative to Earth (in km/s) would be needed?\[ \underline{\hspace{5cm}} \, \text{km/s} \]### In what direction should you leave Earth?- [ ] opposite the direction of Earth’s orbital velocity- [x] towards the Sun- [ ] away from the Sun- [ ] in the direction of Earth’s orbital velocity### (b) What will be the shape of the trajectory?- [ ] a circle- [ ] an ellipse- [x] a hyperbola- [ ] a parabola*Note*: Correct answers are marked with a blue dot. Rectangles containing red "X" marks indicate incorrect choices.---### Explanation#### 1. Direction for Minimum Speed:To escape the solar system with minimum speed, launching your spacecraft **towards the Sun** requires the least velocity because this direction leverages the Sun's gravitational pull to increase speed.#### 2. Trajectory Shape:The shape of the escape trajectory when leaving the gravitational influence of both Earth and the Sun would be a **hyperbola** if the escape speed is beyond the combined gravitational escape velocities of Earth and the Sun.Feel free to freely explore the math and physics behind escape velocities and orbital dynamics through practical examples and simulations available on our website.---

Given information: Here, v0 is the Earth’s orbital speed, RE and RS are the radius of the Earth and…

Consider the process of escaping from the solar system starting from the surface of Earth. Assume there are no other bodies involved. Earth has an orbital speed about the Sun of 29.8 km/s. Hint: 2GM does not apply. Use mv,? GM_m GM m 2 GM-m GMs" which includes the potential energy of both Earth and the Sun. 2.5 V. esc 2,E (a) What minimum speed relative to Earth (in km/s) would be needed? km/s In what direction should you leave Earth? O opposite the direction of Earth's orbital velocity O towards the Sun O away from the Sun O in the direction of Earth's orbital velocity (b) What will be the shape of the trajectory? O a circle O an ellipse O a hyperbola O a parabola

Consider the process of escaping from the solar system starting from the surface of Earth. Assume there are no other bodies involved. Earth has an orbital speed about the Sun of 29.8 km/s. Hint: 2GM does not apply. Use mv,? GM_m GM m 2 GM-m GMs" which includes the potential energy of both Earth and the Sun. 2.5 V. esc 2,E (a) What minimum speed relative to Earth (in km/s) would be needed? km/s In what direction should you leave Earth? O opposite the direction of Earth's orbital velocity O towards the Sun O away from the Sun O in the direction of Earth's orbital velocity (b) What will be the shape of the trajectory? O a circle O an ellipse O a hyperbola O a parabola

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

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# Escape Velocity and Trajectory

Consider the process of escaping from the solar system starting from the surface of Earth. Assume there are no other bodies involved. Earth has an orbital speed about the Sun of 29.8 km/s. Use the following equation which includes the potential energy of both Earth and the Sun:

\[ v_{\text{esc}} = \sqrt{\frac{2GM}{R}} \]

does not apply. Use:

\[ \frac{1}{2} mv_1^2 - \frac{GM_E m}{r_{1,E}} = \frac{1}{2} mv_2^2 - \frac{GM_E m}{r_{2,E}} - \frac{GM_S m}{r_{2,S}} \]

### (a) What minimum speed relative to Earth (in km/s) would be needed?

\[ \underline{\hspace{5cm}} \, \text{km/s} \]

### In what direction should you leave Earth?
- [ ] opposite the direction of Earth’s orbital velocity
- [x] towards the Sun
- [ ] away from the Sun
- [ ] in the direction of Earth’s orbital velocity

### (b) What will be the shape of the trajectory?
- [ ] a circle
- [ ] an ellipse
- [x] a hyperbola
- [ ] a parabola

*Note*: Correct answers are marked with a blue dot. Rectangles containing red "X" marks indicate incorrect choices.

---

### Explanation

#### 1. Direction for Minimum Speed:

To escape the solar system with minimum speed, launching your spacecraft **towards the Sun** requires the least velocity because this direction leverages the Sun's gravitational pull to increase speed.

#### 2. Trajectory Shape:

The shape of the escape trajectory when leaving the gravitational influence of both Earth and the Sun would be a **hyperbola** if the escape speed is beyond the combined gravitational escape velocities of Earth and the Sun.

Feel free to freely explore the math and physics behind escape velocities and orbital dynamics through practical examples and simulations available on our website.

---

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