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Chapter 13: Problem 29

Deimos, a moon of Mars, is about \(12 \mathrm{~km}\) in diameter with mass \(1.5 \times 10^{15} \mathrm{~kg} .\) 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! (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 bascball game?

Short Answer

Expert verified
a) The speed required for the baseball to orbit Deimos is approximately \(145 \, m/s\). This is roughly equivalent to the speed of sound, so it is highly unlikely anyone could throw a baseball at this speed. b) To hit the baseball back, the player would need to wait approximately 5.08 hours. Due to the significant waiting time, this would not result in an action-packed game of baseball.

Step by step solution

01

Calculate the gravitational field strength

The gravitational field strength at the surface of Deimos can be derived from the universal gravitation equation \(F = G*(M*m)/r^2\), where \(F\) is the force of gravity, \(G\) is the gravitational constant (\(6.673 \times 10^{-11} \mathrm{Nm^2/kg^2}\)), \(M\) is the mass of Deimos, \(m\) is the mass of the baseball and \(r\) is the distance between the center of Deimos and the baseball (which would be the radius of Deimos). Because we are interested in the gravitational field strength \(g\), we can eliminate \(m\) because it will cancel out (as the force exerted on the baseball \(F = m*g\)). Therefore, \(g = G*M/r^2\)
02

Calculate the orbiting speed of the baseball

The required speed to keep a body in a circular orbit is given by the formula \(v = \sqrt{g \cdot r}\) where \(v\) is the speed, \(g\) is the gravitational field strength, and \(r\) is the radius of the orbited body.
03

Calculate the time it takes for the baseball to return

The time it will take for the baseball to complete one whole orbit can be derived from the formula \(T = 2\pi*r/v\) where \(T\) is the period (time taken for one whole orbit), \(r\) is the radius of the orbited body and \(v\) is the speed. This will give the time in seconds. To convert this to hours, we need to divide by \(3600\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Field Strength
The gravitational field strength, often denoted as \( g \), is a measure of the force exerted by a body like Deimos on another object such as a baseball. To calculate \( g \) on Deimos, we use the formula: \[ g = \frac{G \times M}{r^2} \] where \( G \) is the gravitational constant \( (6.673 \times 10^{-11} \, \mathrm{Nm^2/kg^2}) \), \( M \) is the mass of Deimos, and \( r \) is the radius of Deimos, which is half of its diameter. This measures the strength of gravity at Deimos鈥� surface.

Because the mass of the baseball cancels out in the formula, gravitational field strength tells us how much force a unit mass would feel at a point in space.

Understanding \( g \) helps us figure out how much force is needed to keep the baseball in orbit and is crucial for solving problems involving orbital mechanics.
Circular Orbit
An object in a circular orbit continuously falls toward the planet it is orbiting but never actually reaches it because it has enough tangential speed to keep missing it. For the baseball to orbit Deimos, it must move at a speed that perfectly balances gravitational pull toward Deimos with its inertia moving tangentially. This speed is given by the formula:

\[ v = \sqrt{g \times r} \]
where \( v \) is the orbital speed, \( g \) is the gravitational field strength, and \( r \) is the radius from the center of Deimos.

This equation helps us understand that at different radii or gravitational strengths, different speeds are required to maintain an orbit. For Deimos, with very weak gravity due to its small size and mass, this speed is significantly less than what is required around larger bodies like Earth.
Physics Problem Solving
Solving physics problems like the one involving Deimos and a baseball requires methodical steps and logical reasoning.

Here are some tips to follow:
  • Identify the known variables, such as masses, radii, and gravitational constants.
  • Use relevant equations systematically to find unknowns, step by step.
  • Check that equations are dimensionally consistent鈥攗nits should match and cancel appropriately.
In our problem, first, we calculated the gravitational field strength \( g \) to find \( v \), the speed for a circular orbit.

Finally, we calculated the time it takes for the baseball to return using the orbit period formula:

\[ T = \frac{2\pi r}{v} \]
Keeping these steps in mind ensures clarity and accuracy, helping you to understand and solve complex physics problems effectively.

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