# Launching a projectile into Orbit

• Noone1982
In summary, the equation for a projectile using cylindrical coordinates is a simple matter of r = 0.5a*t^2. With vectors.
Noone1982
I completely understand how normal projectiles work on a "flat" world using the x and y components.

$$x\; =\; V_{x}t\; +\; x_{o}$$

$$y\; =\; Vyt\; +\; y_{o}\; +\; 0.5gt^{2}$$

I am confused if we place the Earth on the XY plane and launch a projectile at some angle. How are the equations different?

It seems that both Y and X equations should contain a gravity component using sine or cosine to take this into account. If no gravity in the x equations exists, the projectile would never come back to earth.

Noone1982 said:
I completely understand how normal projectiles work on a "flat" world using the x and y components.

$$x\; =\; V_{x}t\; +\; x_{o}$$

$$y\; =\; Vyt\; +\; y_{o}\; +\; 0.5gt^{2}$$

I am confused if we place the Earth on the XY plane and launch a projectile at some angle. How are the equations different?

It seems that both Y and X equations should contain a gravity component using sine or cosine to take this into account. If no gravity in the x equations exists, the projectile would never come back to earth.
I don't quite understand your question. The Earth gravity force always points in one direction only...toward the center of the earth. If you are implying that you are viewing the Earth from say the moon, such that the Earth lies in the XY plane, and a projectile is launched from the equator in a generally left to right direction, horizonatlly or at some angle, then the y-axis can be chosen as the horizontal axis, and the x-axis as the vertical axis. There is no gravity component along the x axis.

These equations were set up by directing the y-axis along the straight down or up direction - that is along a radius line of the earth. So if you work on the scale of the Earth these directions will change during the course of a projectile's flight. We tend to develop/use equations that describe the motion of a projectile in other types of coordinate systems. Anyway, the value of g will decrease as it goes further away from the Earth so these equations are not very helpfull in this context.

How would I set up this problem in cylindrical coordinates?

With R(t) r-hat and phi(t) phi-hat?

It has been quite a while since I have worked on orbital theory and it can get quite involved, but one of the most usefull equations I have found is the relation between the object's angular momentum per unit mass of the object and its angular velocity about the force centre:

$$h=r^2 \omega =\ constant$$

This is due to the fact that the attractive force on the object working only along the line connecting the two. So from this it follows that

$$\dot{\theta} = \frac{h}{r^2}$$

In the radial direction things look similar

$$\ddot{r} = -\frac{GM}{r^2}$$

due to Newton's law of universal gravitational attraction.

Other usefull approaches is energy considerations. Since the object is experiencing a conservative force its energy will remain constant.

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Hmm. I have been going about this another way. My advisor told me it was a simple matter of using essentially R = 0.5a*t^2 but with vectors. Here is what I'v been doing:

http://homepage.mac.com/matthewjacques1/rocket.jpg

Last edited by a moderator:

## 1. How is the launch trajectory calculated for a projectile into orbit?

The launch trajectory is calculated using mathematical equations based on the desired orbit, the mass and speed of the projectile, and the gravitational pull of the planet. Specific calculations include determining the required velocity and angle of launch, as well as accounting for atmospheric drag and other external factors.

## 2. What is the difference between a ballistic and orbital trajectory?

A ballistic trajectory is a curved path that follows the laws of gravity and is influenced by air resistance, but does not enter orbit. An orbital trajectory is a path that allows an object to continuously fall towards the planet while also maintaining a sufficient horizontal velocity to avoid crashing into the surface.

## 3. How does the Earth's rotation affect the launch of a projectile into orbit?

The Earth's rotation plays a crucial role in launching a projectile into orbit. Launching from the equator takes advantage of the Earth's rotational speed, allowing the projectile to gain additional velocity and conserve fuel. Additionally, the orientation of the Earth's rotation must be considered when calculating the launch trajectory.

## 4. What is the most efficient way to launch a projectile into orbit?

The most efficient way to launch a projectile into orbit is through a process called "gravity assist." This involves using the gravitational pull of other planets or moons to gain additional speed and change the trajectory of the projectile. This method reduces the amount of fuel needed for the launch and allows for more precise control of the orbit.

## 5. How do scientists ensure a safe and successful launch into orbit?

Scientists use a variety of measures to ensure a safe and successful launch into orbit. This includes thorough testing and simulations, careful selection and preparation of launch site and equipment, and constant monitoring and adjustments during the launch process. Additionally, extensive research and analysis are conducted beforehand to anticipate and mitigate potential risks and challenges.

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