Solution to Moon Problem 211

Summary

In the video, Keith Norman discusses a problem involving launching a satellite into a circular orbit around the Moon. He explains the necessary calculations for achieving a stable orbit, including determining the circular velocity (1.68 km/s) and period (1.8 hours). Keith also covers the concept of escape velocity, which is 2.38 km/s, and how it relates to the formation of elliptical and hyperbolic orbits. The video provides a clear understanding of orbital mechanics and the factors influencing satellite trajectories.

Takeaways

• 📅 Today's date is Monday, September 16, 2024.
• 🎥 The video is a solution by Keith Norman to a problem, not specified as difficult.
• 🌕 The problem involves launching a satellite into a circular orbit around the Moon.
• 🚀 The satellite is launched from a gun with an adjustable velocity to achieve circular orbit.
• 📐 The gravitational force between the Moon and the satellite must equal the centripetal force for a circular orbit.
• 🔢 For a circular orbit, the calculated velocity is 1.68 km/s, and the period is approximately 1.8 hours or 6,496 seconds.
• 🌐 Part B and Part C of the problem require understanding of escape velocity, derived in lecture 14.
• 💨 The escape velocity, related to the circular velocity by a factor of √2, is calculated to be 2.38 km/s.
• 🛰️ For velocities less than escape velocity, the orbit is bound and elliptical; for velocities above, the orbit is unbound and hyperbolic.
• 📉 As velocity increases beyond the minimum for a stable orbit, the focus of the elliptical orbit moves further away, eventually leading to a parabolic and then hyperbolic path.

Q & A

What is the main topic discussed in the video script?

The main topic discussed in the video script is the solution to Walter Le's problem 2011 by Keith, which involves calculations related to a satellite's orbit around the Moon.

What is the significance of the number 1.68 km/s mentioned in the script?

The number 1.68 km/s is the calculated circular velocity required for a satellite to maintain a stable circular orbit around the Moon.

What is the period of the satellite's orbit as described in the script?

The period of the satellite's orbit is approximately 1.8 hours or 6,496 seconds.

What is the escape velocity from the Moon, as discussed in the script?

The escape velocity from the Moon is calculated to be 2.38 km/s, which is derived by multiplying the circular velocity by the square root of 2.

What type of orbit does the satellite have if it is launched at a velocity of 2 km/s, according to the script?

If the satellite is launched at a velocity of 2 km/s, it will have a bound elliptical orbit.

What happens to the satellite's orbit if it is launched at a velocity greater than the escape velocity?

If the satellite is launched at a velocity greater than the escape velocity, it will have an unbound hyperbolic orbit and will not return to the Moon.

What is the minimum velocity required for a stable orbit around the Moon, as per the script?

The minimum velocity required for a stable orbit around the Moon is 1.68 km/s.

What is the relationship between the gravitational force and the centripetal force for a satellite in a circular orbit, as explained in the script?

For a satellite in a circular orbit, the gravitational force between the satellite and the Moon must match the centripetal force required for circular motion.

What is the role of the gravitational constant in calculating the satellite's orbit, according to the script?

The gravitational constant is used in the calculations to determine the force of gravity between the satellite and the Moon, which is essential for determining the satellite's orbit.

Why does the script mention that being a few meters above the Moon's surface does not significantly affect the calculations?

The script mentions that being a few meters above the Moon's surface does not significantly affect the calculations because the Moon's radius is approximately 1.7 million meters, making a small height difference negligible.

What is the focus of an elliptical orbit, and how does it relate to the satellite's velocity, as discussed in the script?

In an elliptical orbit, one focus remains at the center of the Moon, and the other focus moves away as the satellite's velocity increases. At a certain velocity, the second focus moves to infinity, indicating a parabolic path.

Outlines

00:00

🌕 Introduction to Lunar Orbital Mechanics

The speaker begins by setting the context of the video, mentioning the date and introducing the topic to be discussed: a problem-solving approach by Keith Norman. The problem revolves around the concept of launching a satellite into a circular orbit around the Moon. The speaker highlights that there are many correct solutions, and Keith's approach is particularly appreciated for its unique twist. The video is part of a lecture series, specifically lecture 14, which covers the basics of orbital mechanics, including the gravitational force and centripetal force required for a stable circular orbit. The speaker uses the example of launching a satellite from a gun at a velocity that can be adjusted to achieve this orbit. The gravitational force between the Moon and the satellite is equated to the centripetal force needed for circular motion, leading to an expression that can be solved using known values for the Moon's mass and radius, as well as the gravitational constant. The resulting velocity for a stable circular orbit is calculated to be 1.68 km/s, and the period of the orbit is determined to be approximately 1.8 hours or 6,496 seconds.

05:00

🚀 Escaping Lunar Gravity and Orbital Dynamics

In the second paragraph, the speaker delves into the concepts of escape velocity and the dynamics of elliptical and hyperbolic orbits. The escape velocity, which is the minimum velocity needed to break free from the Moon's gravitational pull, is derived from the circular velocity and is found to be 2 km/s. The speaker explains that if a satellite is launched at a velocity less than the escape velocity but more than the circular velocity, it will follow an elliptical orbit. As the launch velocity increases, the orbit's focus moves further away from the Moon until it reaches the escape velocity, at which point the orbit becomes parabolic. Beyond the escape velocity, the satellite enters a hyperbolic orbit, meaning it will never return to the Moon. The speaker visually illustrates these concepts, showing the progression from a stable orbit to an elliptical, parabolic, and finally hyperbolic path as the launch velocity increases. The summary concludes with a clear explanation of the different types of orbits and their corresponding velocities.

Keywords

💡Circular Orbit

A circular orbit is a path followed by an object in space where the object's distance from the center of the orbit remains constant. In the video, the concept is used to describe the desired trajectory for a satellite launched from a hypothetical gun on the moon. The speaker explains how to achieve a circular orbit by matching the gravitational force with the centripetal force required for circular motion.

💡Centripetal Force

Centripetal force is the force that keeps an object in circular motion, always directed towards the center of the circle. In the context of the video, the speaker discusses how this force is necessary for maintaining a satellite in a circular orbit around the moon, ensuring it doesn't fly off into an elliptical or hyperbolic path.

💡Gravitational Force

Gravitational force is the attractive force that exists between any objects with mass. In the video, the speaker calculates the gravitational force between the moon and a satellite to determine the necessary velocity for a stable circular orbit. This force is crucial for keeping the satellite in orbit.

💡Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from the gravitational influence of a celestial body without further propulsion. The video discusses how the escape velocity is calculated and how it relates to the velocity needed for a circular orbit around the moon. The speaker uses the escape velocity to determine the fate of a satellite depending on its launch speed.

💡Elliptic Orbits

An elliptic orbit is a type of orbit where the path of an object is elliptical, with one focus being the center of the celestial body it orbits. The video explains that if a satellite is launched at a velocity greater than the minimum for a stable orbit but less than the escape velocity, it will follow an elliptic orbit around the moon.

💡Hyperbolic Orbit

A hyperbolic orbit is an open orbit where an object passes by a celestial body and then moves away indefinitely, never to return. In the video, the speaker describes that if a satellite is launched at a velocity greater than the escape velocity, it will follow a hyperbolic orbit and not return to the moon.

💡Lecture 14

Lecture 14 is referenced in the video as a source where the principles of circular orbits and the necessary calculations are explained in detail. It serves as a foundational resource for understanding the concepts discussed in the video, such as the relationship between gravitational force and centripetal force.

💡Velocity

Velocity in the video refers to the speed of the satellite in orbit around the moon. The speaker calculates the velocity needed for different types of orbits, such as circular, elliptical, and hyperbolic. Velocity is a key factor in determining the satellite's trajectory and whether it will remain in orbit or escape the moon's gravity.

💡Stable Orbit

A stable orbit is one where the forces acting on an object are balanced, allowing it to maintain a consistent path without uncontrolled deviation. The video describes the conditions for achieving a stable orbit around the moon, such as launching the satellite at the correct velocity to counteract the moon's gravity.

💡Parabolic Path

A parabolic path is a specific type of trajectory that an object can take when it has just enough velocity to escape a celestial body's gravitational pull but not enough to move away indefinitely. The video uses the parabolic path to illustrate the threshold between a bound orbit (elliptical) and an unbound orbit (hyperbolic).

💡Walter Le's Problem 2011

Walter Le's Problem 2011 is the main subject of the video, a hypothetical scenario that involves calculating the necessary velocities for different types of orbits around the moon. The problem is used to explore concepts like gravitational force, escape velocity, and the physics of orbital mechanics.

Highlights

Introduction to the video discussing a solution by Keith Norman to a problem from Walter Le's lecture.

The problem involves calculating the circular orbit of a satellite around the moon.

Keith Norman's solution is described as always providing an extra twist.

The video references lecture 14 of 801 and lecture 22 for details on elliptic orbits.

The importance of matching gravitational force with centripetal force for circular motion is emphasized.

The calculation for the velocity required for a circular orbit is explained.

The gravitational constant is mentioned as a necessary value for the calculations.

The calculated circular velocity for a circular orbit around the moon is 1.68 km/s.

The period of the orbit, calculated as the circumference of the moon divided by velocity, is approximately 1.8 hours.

The concept of escape velocity is introduced for part C of the problem.

Escape velocity is calculated as the circular velocity multiplied by the square root of 2.

The calculated escape velocity is 2 km/s.

A discussion on the types of orbits based on velocity: bound (elliptical) and unbound (hyperbolic).

Explanation of how increasing velocity affects the shape of the orbit from circular to elliptical.

Description of the transition from elliptical to parabolic orbit at escape velocity.

The final part of the video discusses the satellite's path for velocities above escape velocity, resulting in hyperbolic orbits.

Summary of the different orbits and velocities discussed in the video.

The video concludes with a thank you note from the presenter.

Transcripts

hello hello

hello today is

Monday September 16

2024 here

follow the video

Solution by Keith

Norman it was not a difficult

problem I didn't count how many correct

an there were but there were many more

than

20 but Keith solution is always very

nice and he often gives it an extra

twist which he also does this

time

so Walking on the Moon yes that will be

fun and if you're ready for Kei

solution I am

this is Keith's solution to Walter Le's

problem

2011 uh and it's completely covered in

the first part of lecture 14 of 801 and

uh lecture 22 is all about uh elliptic

orbits which we will come on to in a

moment so um first of all we're dealing

with a circular orbit we have the moon

and we have a satellite that we're going

to launch out of a gun with a with a a

velocity that we can we can adjust

adjust so we'll pick a velocity that

require that gives us a circular orbit

so if we're launching say 10 m above um

the the moon and and R here is 1.7

million uh met or so so being a few

meters above is isn't going to affect

this in any significant amount um okay

so we want a circular orbit and for that

to

happen we need the force between of

between the little mass of the satellite

the moon which is the gravitational

force here this this term must match the

centripetal force required for circular

motion totally covered in lecture 14 so

we need to match these two terms so we

do that we get this

expression we're given uh M and R in the

problem we can look up the gravitational

constant and we can feed in the numbers

for part A we get that uh circular U

velocity for a circular orbit uh turns

out to be 1.68

km/s uh similarly the similarly the

period um is going to be the uh

circumference here of the Moon um

divided by the velocity and I get this

which turns out to be 1.8 hours or 6,496

seconds or thereabouts so that's part A

and Part B

for part C we and Part D we need to know

something about the escape Velocity um

again in lecture 14 uh Walter derives an

expression for the escape Velocity I

won't do it here if you if you take the

trouble you will find how to do it uh in

his lecture 14 in the first 15 minutes

or so um and he notes that the escape

Velocity is related to the the circular

um the velocity for a circular orbit

which we've just calculated it's related

by a factor of root2 so to find escape

velocity I simply take the previous

answer and multiply byun2 and I get that

okay that's escape velocity for uh part

C the velocity is 2 km a second clearly

less than escape velocity so I get a

bound orbit which will be

elliptical uh for Part D I'm above the

escape Velocity so the orbit is Unbound

in other words the satellite will never

return to the Moon uh and it is a

hyperbolic

orbit and just to summarize and I hope

this doesn't confuse people what we

have is again nothing is to scale we've

got we're here's the moon um we're

launching our satlite um initially we

launch with the minimum uh velocity for

a stable orbit which is this here 1.68

km a

second anything less and we we would

simply fall onto the moon's surface uh

then as we start to increase the

velocity say we say we we make it I

don't know 1 Point 1.75 or something

we'd start to get an ellipse forming out

here somewhere and getting bigger and

bigger in this in this whole region um

and what's happening is that one of the

the focus of the ellipse one one Focus

stays here and the other one starts to

move away so at some velocity V = 2 km a

second we'll have the focus uh that

we're orbiting about remaining at the

center of the Moon and there will be a

second one out in space along this line

so I've shown here the stable orbit we

get for V = 2 then as we keep going

we're going to still get elipses that

this Focus will fly off to infinity and

we'll reach a point where we have a

parabolic path

uh with the escape velocity of V = 2.38

beyond that nothing will ever return and

we get our hyperbolic paths and hence

for V = 3 we're out here somewhere Part

D and that is my answer thank you