Problem 58 - University Physics with Modern Physics

Question

Your starship, the Aimless Wanderer, lands on the mysterious planet Mongo. As chief scientist-engineer, you make the following measurements: \(A 2.50-\) -kg stone thrown upward from the ground at 12.0 \(\mathrm{m} / \mathrm{s}\) returns to the ground in 6.00 s; the circumference of Mongo at the equator is \(2.00 \times 10^{5} \mathrm{km} ;\) and there is no appreciable atmosphere on Mongo. The starship commander, Captain Confusion, asks for the following information: (a) What is the mass of Mongo? (b) If the Aimless Wanderer goes into a circular orbit \(30,000 \mathrm{km}\) above the surface of Mongo, how many hours will it take the ship to complete one orbit?

Step-by-Step Solution

Answer

(a) The mass of Mongo is calculated using the gravitational acceleration and its radius. (b) The Aimless Wanderer takes several hours to complete one orbit, calculated from the orbital period formula.

Step 1: Calculate the acceleration due to gravity on Mongo

The stone is thrown upwards at a speed of 12.0 m/s and takes 6.00 s to return to the ground. Hence, the time to reach the highest point is half of the total time, i.e., 3.00 s. The acceleration can be calculated using the equation of motion: \[ v = u + at \]where \( v = 0 \) m/s (as the stone stops momentarily at the highest point), \( u = 12.0 \) m/s and \( t = 3.00 \) s. Solving for \( a \):\[ 0 = 12.0 + a \times 3.00 \]\[ a = \frac{-12.0}{3.00} = -4.00 \text{ m/s}^2 \]The negative sign indicates that the acceleration is directed downwards. So, the acceleration due to gravity on Mongo is \( 4.00 \text{ m/s}^2 \).

Step 2: Calculate the mass of Mongo

Use the formula for gravitational acceleration:\[ g = \frac{G M}{R^2} \]Where \( g = 4.00 \text{ m/s}^2 \), \( R = \frac{C}{2\pi} = \frac{2.00 \times 10^5 \times 10^3}{2\pi} \text{ m} \) (circumference \( C \)), and \( G = 6.674 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2} \).Rearrange to solve for \( M \):\[ M = \frac{g R^2}{G} \]Substitute the values and calculate \( M \).

Step 3: Calculate the orbital radius

The ship orbits 30,000 km above the surface, so the orbital radius \( R_o \) is:\[ R_o = R + 30000 \times 10^3 \text{ m} \]

Step 4: Calculate the orbital period

Use Kepler's Third Law for orbital motion:\[ T^2 = \frac{4\pi^2 R_o^3}{G M} \]Where \( T \) is the orbital period. Rearrange to solve for \( T \):\[ T = 2\pi \sqrt{\frac{R_o^3}{G M}} \]Substitute the previously calculated \( M \), \( R_o \), and \( G \). Compute \( T \) in seconds and convert this to hours by dividing by 3600. Round to the nearest hour.

Key Concepts

Gravitational Acceleration

Gravitational acceleration is the rate at which objects accelerate towards the surface of a celestial body due to gravity. On Earth, this rate is approximately 9.81 m/s². However, on different planets, this figure can vary depending on the planet's mass and radius.

When dropping objects or throwing them upwards, they accelerate towards the surface due to the gravitational pull. For instance, on the planet Mongo, we calculated this acceleration as 4.00 m/s² using the formula:\[ v = u + at\]Here, the initial velocity \( u \) was 12.0 m/s, but the stone stops momentarily at its peak, making \( v \) equal to 0. Solving this equation helps us determine the gravitational pull on Mongo.

It's crucial in physics to understand that gravitational acceleration affects everything equally, regardless of the mass of the object being dropped or thrown. This concept is central to understanding how things move on planets beyond Earth.

Orbital Mechanics

Orbital mechanics is the science of how celestial bodies move through space. It involves understanding how planets, moons, and spacecraft move around larger bodies due to gravity. This field is vital for spacecraft navigation and understanding celestial motion.

When calculating the orbit of a spacecraft like the Aimless Wanderer around Mongo, you need to find the orbital radius, which is the sum of the planetary radius and the height of the orbit above the planet's surface. This ensures safe and predictable orbiting paths.

Once the radius is determined, it becomes easier to handle different computations related to the orbit, such as determining speed or time taken for one complete orbit. This knowledge is essential for space missions and sat navigation.

Kepler's Third Law

Kepler's Third Law is a key principle in orbital mechanics, stating that the square of the orbital period \( T \) of a planet is proportional to the cube of the semi-major axis \( R_o \) of its orbit. This relationship can be expressed with the formula:\[ T^2 = \frac{4\pi^2 R_o^3}{G M}\]Where \( G \) is the gravitational constant, and \( M \) is the mass of the central body.

This law helps us calculate how long it takes for a spaceship like the Aimless Wanderer to complete one full orbit around Mongo. By rearranging the formula to solve for \( T \), you can find the time it takes for one orbit, converting the result to a usable unit like hours.

Kepler's Third Law applies to any object in orbit, illustrating a beautiful predictability and elegance to the motion of planets and satellites.

Motion Equations

Motion equations are mathematical tools used to describe the movement of objects. These equations allow us to predict how an object will move over time given its initial conditions. They include key components such as velocity, acceleration, displacement, and time.

In the problem about Mongo, we use motion equations to calculate the gravitational acceleration by analyzing the stone's motion: \[ v = u + at\]Where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.

These equations are not just limited to vertical motion but apply to any type of uniform motion. They're fundamental in physics, allowing us to solve problems related to displacement, speed, and changes in velocity over time.

Understanding how to manipulate these equations can make complex physics problems much simpler and provide a clearer insight into the physical world around us.