Problem 41 - College Physics

Question

Baseball on Deimos! Deimos! Deimos, a moon of Mars, is about 12 \(\mathrm{km}\) in diameter, with a mass of \(2.0 \times 10^{15} \mathrm{kg}\) . Suppose you are stranded alone on Deimos and want to play a one-person game of baseball. You would be the pitcher, and you would be the batter! With what speed would you have to throw a baseball so that it would go into orbit and return to you so you could hit it? Do you think you could actually throw it at that speed?

Step-by-Step Solution

Answer

The orbital speed is approximately \\( 4.71 \, \text{m/s} \\. Yes, you could throw a baseball this fast.

Step 1: Understand Orbital Velocity

To solve this problem, we need to calculate the orbital velocity required for a baseball to orbit around Deimos. The formula for orbital velocity \( v \) is given by \( v = \sqrt{\frac{G \, M}{r}} \,\) where \( G = 6.67 \times 10^{-11} \, \text{N(m/kg)}^2\) is the gravitational constant, \( M \) is the mass of the celestial body (Deimos), and \( r \) is the radius of the orbit.

Step 2: Determine the Known Values

We know the mass of Deimos is \( M = 2.0 \times 10^{15} \, \text{kg} \). Deimos has a diameter of \( 12 \, \text{km} \,\) so its radius \( r \) is half of that, which is \( 6 \, \text{km} = 6,000 \, \text{m} \).

Step 3: Calculate Orbital Velocity

Substitute the known values into the formula for orbital velocity: \[ v = \sqrt{\frac{6.67 \times 10^{-11} \, \text{N(m/kg)}^2 \times 2.0 \times 10^{15} \, \text{kg}}{6,000 \, \text{m}}} \]. Simplifying inside the square root gives \( v = \sqrt{\frac{1.334 \times 10^5}{6,000}} = \sqrt{22.2333} \,\) yielding \( v \approx 4.71 \, \text{m/s} \).

Step 4: Consider Practical Implications

A speed of \( 4.71 \, \text{m/s} \) is quite slow for throwing a baseball (compared to typical speeds upwards of 40 \, \text{m/s} or more on Earth). Thus, in practice, you could easily achieve this speed and indeed throw a baseball into orbit around Deimos.

Key Concepts

Gravitational Constant

The Gravitational Constant, often symbolized as \( G \), is a crucial constant in physics, representing the force of attraction between two objects with mass. - It is approximately \( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \).
- This value may seem tiny, and that's because gravitational attraction is generally weak unless the masses involved are significantly large, like celestial bodies.
In the context of calculating orbital velocity, \( G \) ensures that the calculations take into account the correct scale of gravitational influence between a moon or planet and an object attempting to orbit it. Understanding and using \( G \) correctly is fundamental in orbital mechanics and astronomy.

Mass of Celestial Bodies

The mass of celestial bodies, such as planets, moons, and stars, plays an essential role in determining gravitational pull. - Deimos, Mars' smaller moon, has a mass of \( 2.0 \times 10^{15} \, \text{kg} \).
- This mass affects how much gravity it can exert on objects, like a baseball, trying to enter orbit around it.
Since gravitational force is directly proportional to the mass, more massive bodies exert a stronger gravitational pull, influencing the necessary velocity an object needs to achieve to remain in orbit. Knowing the mass helps in solving various physics problems related to space, enabling accurate predictions in orbital mechanics.

Orbital Mechanics

Orbital Mechanics is the field of physics that studies the motions of objects in space, governed mainly by the laws of gravity. It's a fascinating science involving the calculation of trajectories, including how objects like a baseball thrown on Deimos can orbit the moon. - The formula for orbital velocity \( v = \sqrt{\frac{G \cdot M}{r}} \) gives us the speed needed for an object to remain in a stable, circular orbit.
- As seen with the Deimos baseball example, a surprisingly low speed of \( 4.71 \, \text{m/s} \) is all that's needed, highlighting unique conditions due to Deimos' small mass and size.
Orbital mechanics requires understanding of gravitational forces, mass, and radius, providing crucial insights into space missions and satellite deployment.

Physics Problems

Physics problems frequently incorporate real-world applications, such as calculating the speed needed for an object to orbit a celestial body, as demonstrated with the baseball and Deimos. - These problems sharpen problem-solving skills and understanding of physics principles.
- Solutions often demand calculations using constants, known values of mass, radius, and gravitational forces.
Breaking down physics problems into step-by-step solutions, like computing the necessary speed for an orbit, helps in grasping abstract concepts, making them manageable and comprehensible.

Astronomical Calculations

Astronomical calculations are vital for understanding and predicting celestial events and behaviors. They involve math and physics to calculate distances, velocities, and forces in the universe. - For example, determining how fast a baseball must travel to orbit Deimos involves math and physics principles, offering a practical illustration of these calculations.
- Such calculations are fundamental for spacecraft missions, predicting orbits, and understanding the dynamics of the solar system.
By mastering these calculations, scientists can navigate and explore the vastness of space, enhancing our understanding of the cosmos and the forces that govern it.