Problem 25 - University Physics with Modern Physics

Question

Deimos, a moon of Mars, is about 12 km in diameter with mass 1.5 \(\times\) 10\(^{15}\) 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! (a) With what speed would you have to throw a baseball so that it would go into a circular orbit just above the surface and return to you so you could hit it? Do you think you could actually throw it at this speed? (b) How long (in hours) after throwing the ball should you be ready to hit it? Would this be an action-packed baseball game?

Step-by-Step Solution

Answer

To put a baseball into orbit on Deimos requires a speed of 129.18 m/s. This is not physically achievable by a human throw. The ball returns after about 4.87 minutes.

Step 1: Calculating Gravitational Force

To find the speed needed for a circular orbit, we first use the formula for gravitational force: \[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]Here, \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2} \), \( m_1 \) is the mass of Deimos \( 1.5 \times 10^{15} \text{ kg} \), \( m_2 \) is the mass of a baseball (which cancels out later), and \( r \) is the radius (\( \frac{12}{2} = 6\) km, or \( 6000 \) m).

Step 2: Using Circular Motion Equation

For a circular orbit, the centripetal force needed equals the gravitational force:\[ \frac{m_2 \cdot v^2}{r} = \frac{G \cdot m_1 \cdot m_2}{r^2} \]Simplifying gives:\[ v^2 = \frac{G \cdot m_1}{r} \]\[ v = \sqrt{\frac{G \cdot m_1}{r}} \]

Step 3: Calculating Velocity

Substitute the known values into the circular motion equation:\[ v = \sqrt{\frac{6.674 \times 10^{-11} \cdot 1.5 \times 10^{15}}{6000}} \]Simplifying gives:\[ v \approx \sqrt{1.6685 \times 10^4} \approx 129.18 \text{ m/s} \]

Step 4: Determining Feasibility of Throw

Consider human limitations; the world record for a baseball pitch is around 45 m/s. It is not feasible for a human to throw at the required 129.18 m/s.

Step 5: Calculating Orbital Period

The orbital period (\( T \)) of the baseball can be found using:\[ T = \frac{2\pi r}{v} \]\[ T = \frac{2\pi \times 6000}{129.18} \approx 292.26 \text{ seconds} \]Convert seconds into hours:\[ T \approx \frac{292.26}{3600} \approx 0.081 \text{ hours} \]

Step 6: Evaluating the Nature of the Game

The baseball returns after about 0.081 hours, or roughly 4.87 minutes. This relatively long duration for a game action would not make the game action-packed.

Key Concepts

Circular Orbit

A circular orbit refers to the motion of an object around another object in a path that maintains a consistent distance from that object. The defining feature of a circular orbit is its constant radius. Imagine a baseball that you pitch and it travels around Deimos (a moon of Mars) in a perfect circle. For such an orbit to be possible, the force that is pulling the baseball towards the center of Deimos must perfectly balance the baseball's inertia that wants to fling it in a straight line. This delicate balance ensures that the baseball will continue to circle Deimos rather than spiraling inward or veering off into space.

To achieve a circular orbit, the velocity must be just right—too fast, and the baseball would drift away; too slow, and it would plunge to the surface.

Gravitational Force

Gravitational force is the attractive force between two masses. Isaac Newton's law of universal gravitation tells us that this force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. Simply put, the bigger the masses are, the stronger their pull toward each other. But as they get farther apart, the force weakens.

For this exercise, we focus on the gravitational pull between Deimos and a baseball. The relevant equation is:

• \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \)

Here, \( G \) is the gravitational constant, \( m_1 \) is the mass of Deimos, \( m_2 \) is the mass of the baseball, and \( r \) is the radius of Deimos. The gravitational force provides the necessary centripetal force to keep the baseball in orbit.

Centripetal Force

Centripetal force is what keeps an object moving in a circular path. "Centripetal" means "center-seeking," a perfect descriptor because this force always points toward the center of the circle that the object is traveling around. For our baseball, this centripetal force is provided by the gravitational attraction between the moon, Deimos, and the baseball itself.

When looking at the forces acting on the baseball in a circular orbit, the equation is balanced like this:

• \( \frac{m_2 \cdot v^2}{r} = \frac{G \cdot m_1 \cdot m_2}{r^2} \)

Notice how the mass of the baseball, \( m_2 \), cancels out, simplifying the calculation. This means that the orbital speed doesn't depend on the mass of the object in orbit—in this case, the baseball.

Orbital Period

The orbital period refers to the time it takes for an object to make one complete orbit around another object. In this case, it's how long the baseball takes to travel around Deimos and return to the starting point. The formula to determine the orbital period \( T \) is:

• \( T = \frac{2\pi r}{v} \)

This formula is derived from the simple concept of speed being distance traveled over time. The circumference of the circle \( (2\pi r) \) is divided by the constant orbital velocity \( v \) to find \( T \).

For Deimos, with an orbital speed of roughly 129.18 m/s and a radius of 6000 m, we find the orbital period to be about 292 seconds, which converts to about 4.87 minutes.

Velocity Calculation

Calculating the velocity needed for a circular orbit around Deimos uses a straightforward relationship between gravitational and centripetal forces. We aim to find a precise speed that balances these forces perfectly. The formula for orbital speed \( v \) is derived as:

• \( v = \sqrt{\frac{G \cdot m_1}{r}} \)

It emerges from equating gravitational force to necessary centripetal force, as detailed in the previous sections.

By plugging in Deimos' mass and radius, along with the gravitational constant, we calculated the velocity to be 129.18 m/s. This precise velocity ensures that a baseball would maintain its circular path around Deimos, should such a throw be possible. However, this speed is significantly higher than what a human could achieve with a baseball throw, making the scenario purely theoretical and an exciting thought experiment.