Question

A reconnaissance spacecraft circles the Moon at very low altitude. Calculate (a) its speed and \((b)\) its period of revolution. Take needed data for the Moon from Appendix C.

Short answer

The spacecraft's speed and period of revolution around the moon can be calculated using the laws of circular motion in combination with the gravitational force. The exact numerical values would depend on the values for the moon's mass, the moon's radius and the gravitational constant given in Appendix C.

Step-by-step solution

Calculation for speed

The speed, \(v\), of an object moving in a circle of radius \(r\) under the influence of a gravitational body of mass \(m\) is given by the equation \(v = \sqrt{G \cdot \frac{m}{r}}\) where G is the gravitational constant. From Appendix C, get the required values for r (radius of the moon plus altitude of spacecraft which is very low in this case so it can be approximated to the moon's radius), m (mass of moon), and G (gravitational constant). Plug these values into the equation to get the speed.

Calculation for period of revolution

The period of revolution, \(T\), around a gravitational body of mass \(m\) in a circle of radius \(r\) is given by the equation \(T = \frac{2\pi r}{v}\). With the already calculated speed, \(v\), and the radius, \(r\), plugged into this equation, the student can find the period of revolution.

Key concepts

Gravitational Constant

The Gravitational Constant, often symbolized by the letter "G," plays a fundamental role in calculations related to orbital mechanics. It represents the attractive force between two masses separated by a certain distance.

• The value of "G" is approximately 6.674 × 10^{-11} N(m/kg)^2.
• This constant is crucial when calculating forces and speeds related to celestial objects.

Understanding how "G" affects gravitational forces allows us to determine the speed and period of orbit for a spacecraft. In our specific problem, this constant is essential in calculating the speed at which a spacecraft orbits the Moon.
By substituting the gravitational constant along with the mass of the Moon into relevant formulas, we can understand its role in generating the centripetal force needed to keep the spacecraft in orbit.

Reconnaissance Spacecraft

A reconnaissance spacecraft is a type of satellite or space probe designed for the specific purpose of collecting data and observations about a planet or celestial body.
These spacecraft typically operate at low altitudes, especially over moons or small planets, so they can gather detailed information.

• They equip scientific instruments to examine surface features.
• Their mission can sometimes include mapping, imaging, and even communication tasks.

In the context of orbiting the Moon, such a spacecraft needs to maintain a precise speed and path.
This ensures that it stays in a stable orbit while gathering the necessary observations. The calculations for speed and period of revolution help in planning the spacecraft's mission effectively.

Period of Revolution

The period of revolution refers to the time it takes for a celestial object or spacecraft to complete one full orbit around another body. For our reconnaissance spacecraft around the Moon, understanding this period is critical.
The time for one complete orbit depends on both the orbit's radius and the speed of the spacecraft.

• To calculate the period (T), one can use the formula: T = \frac{2\pi r}{v}.
• Here, \( r \) is the radius of the orbit and \( v \) is the speed calculated previously.

This concept demonstrates how the balance of gravitational interactions and velocity determines an object's orbital period.
In practical applications, knowing the period allows mission planners to time satellite passes over specific areas to the second, enhancing their data collection accuracy.

Speed Calculation

Calculating the speed of a spacecraft orbiting a celestial body like the Moon involves using a specific formula derived from orbital mechanics. This is necessary to ensure that the spacecraft both maintains its orbit and avoids deviation due to gravitational forces.

• The formula used is \( v = \sqrt{G \cdot \frac{m}{r}} \), where:
  • \( v \) represents speed.
  • \( G \) is the gravitational constant.
  • \( m \) is the mass of the celestial body being orbited—in this case, the Moon.
  • \( r \) is the radius of the orbit, often equivalent to the radius of the Moon for very low altitudes.

By substituting the known values into this equation, one can calculate the spacecraft's speed.
A correct speed calculation is crucial. It ensures the spacecraft makes use of gravitational forces effectively to maintain its orbit without burning excess fuel.