Question

The average person can jump off the ground with a velocity of \(7 \mathrm{mph}\) without fear of leaving the planet. However, if an astronaut jumps with this velocity while standing on Halley's Comet, will the astronaut ever come back down? Create a program that allows the user to input a launch velocity (in mph) from the surface of Halley's Comet and determine whether a jumper will return to the surface. If not, the program should calculate how much more massive the comet must be in order to retum the jumper to the surface. Hint: Escape velocity is \(v_{\text {ecape }}=\sqrt{2 \frac{G M}{R}}\), where \(G=6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}\) is the gravitational constant, \(M\) is the mass of the heavenly body, and \(R\) is its radius. Halley's comet has a mass of \(2.2 \times 10^{14} \mathrm{~kg}\) and a diameter of \(9.4 \mathrm{~km}\).

Short answer

The astronaut will not return. The comet needs a mass increase of \(3.438 \times 10^{19} \text{ kg}\) for the astronaut to return.

Step-by-step solution

Convert Velocity

First, convert the velocity from miles per hour (mph) to meters per second (m/s). Use the conversion factor: \(1 ext{ mph} = 0.44704 ext{ m/s}\). Thus, \(7 ext{ mph} = 7 \times 0.44704 ext{ m/s} = 3.12928 ext{ m/s}\).

Calculate Radius

Calculate the radius of Halley's Comet from its diameter. The given diameter is \(9.4\) km, so the radius \(R = \frac{9.4}{2} \text{ km} = 4.7 \text{ km} = 4700 \text{ m}\).

Calculate Escape Velocity

Use the escape velocity formula \(v_{escape} = \sqrt{2 \frac{GM}{R}}\). Substitute \(G = 6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2\), \(M = 2.2 \times 10^{14} \text{ kg}\), and \(R = 4700 \text{ m}\). Calculate to find the escape velocity: \(v_{escape} = \sqrt{2 \frac{6.67 \times 10^{-11} \times 2.2 \times 10^{14}}{4700}} = 0.00572 \text{ m/s}\).

Compare Velocities

Compare the jump velocity of \(3.12928 \text{ m/s}\) to the escape velocity of \(0.00572 \text{ m/s}\). Since the jump velocity is greater than the escape velocity, the astronaut will not return.

Calculate Mass Adjustment

To determine how much more massive the comet must be, set the jump velocity equal to the desired escape velocity and solve for the new mass. Rearranging the escape velocity formula gives us: \(M_{new} = \frac{v^2_{jump} \cdot R}{2G}\). Substituting the known values: \(M_{new} = \frac{(3.12928)^2 \cdot 4700}{2 \times 6.67 \times 10^{-11}}\). Calculating this gives \(M_{new} = 3.438 \times 10^{19} \text{ kg}\).

Determine Mass Increase

Finally, calculate how much more massive the comet must be by finding the difference between the new mass and its original mass: \(\Delta M = M_{new} - M = 3.438 \times 10^{19} \text{ kg} - 2.2 \times 10^{14} \text{ kg}\). Therefore, \(\Delta M = 3.438 \times 10^{19} \text{ kg}\).

Key concepts

Physics in Space Exploration

Space exploration is a fascinating aspect of physics that examines how we can travel and operate beyond Earth. This involves understanding the physical principles that govern movement and forces in space. Escape velocity plays a vital role here, as it's the speed needed to break free from a celestial body's gravitational pull.
The escape velocity is calculated using the formula \(v_{\text{escape}}=\sqrt{2 \frac{G M}{R}}\). This formula shows the importance of a body's mass \(M\) and radius \(R\) in determining the energy required to leave its surface. Halley's Comet, being smaller and less massive than Earth, requires much less velocity to escape. This concept applies not only to asteroids and comets but also to planets and moons in our solar system.
Understanding these fundamentals is crucial for successful space missions, ensuring that spacecraft and astronauts can safely travel to and return from space.

Gravitational Constants

The gravitational constant \(G\) is a fundamental cornerstone of astrophysics and physics as a whole. It describes the force of attraction between any two masses. In our equation for escape velocity, \(G\) helps us understand how strongly two bodies pull on each other.
Its value, \(6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}\), indicates that this force is actually quite small unless at least one of the bodies has a very large mass. This constant is universal, meaning it holds true throughout the universe. Understanding \(G\) allows us to calculate how galaxies hold together, how stars maintain their orbit, and even how astronauts can experience weightlessness in orbit.
For students diving into physics, mastering the use of \(G\) is fundamental to solving many problems in gravitational physics.

Astrophysics Calculations

Astrophysics involves making calculations that explain the behavior of celestial bodies and the forces acting upon them. To solve the escape velocity problem for Halley's Comet, several key calculations were necessary:

• Converting units from miles per hour to meters per second is essential since physics often operates in metric units.
• Calculating the radius from the given diameter helps in parameterizing equations properly.
• Using the escape velocity formula integrates all known variables, linking gravitational strength, mass, and distance.

These calculations illustrate how scientific principles are applied in both theoretical and practical contexts. Whether it's ensuring that a spacecraft can break free from Earth or understanding how a comet orbits the Sun, such calculations are pivotal in space sciences.

Programming in Science Education

Computational programming is an invaluable skill in modern science education, particularly in fields such as physics and astrophysics. In our escape velocity scenario, programming can simplify repetitive or complex calculations, allowing students to simulate and analyze physical concepts effectively.
By writing a program that takes various inputs for velocity or celestial body characteristics, students can quickly determine outcomes like escape velocity.

• Programming allows for automation of calculations, reducing human error.
• It provides a way to visualize data, which can enhance understanding.
• Students can learn to model physical systems and predict changes when variables are adjusted.

This approach equips students with both a deeper understanding of the material and the technical skills needed for modern scientific research and exploration. Embracing programming in science education prepares learners for a world increasingly driven by technology and data analysis.