# A rocket burns out at an altitude h above the Earth's surface

• Benjamin Fogiel
In summary, the conversation is about determining the speed of a rocket very far from Earth after burning out at a certain altitude. The equation used is v=(v02-v2esc)1/2 and the values of kinetic and potential energies are plugged in to simplify the equation and determine the final speed. The concept of escape velocity is also mentioned, which is the value needed to escape Earth's gravitational pull. The final answer is found by setting the final potential energy to 0 at "very far away" distance.
Benjamin Fogiel

## Homework Statement

A rocket burns out at an altitude h above the Earth's surface. Its speed v0 at burnout exceeds the escape speed vesc appropriate to the burnout altitude. Show that the speed v of the rocket very far from the Earth is given by v=(v02-v2esc)1/2

KEf-KEi=Ui-Uf

## The Attempt at a Solution

I plugged in the values of kinetic and potential energies:

(m1v2)/2 - (m1v02)/2 = (-Gm1m2)/r1 + (Gm1m2)/r2

Simplified to get:

v2=(-2Gm2)/r1 + (2Gm2)/r2 + v02

Im not sure where to go from here

What are ##r_1## and ##r_2##?

Escape velocity for mass ##m_1## to escape mass ##m_2## starting at distance ##r## is the value such that ##(1/2)m_1v_0^2 = G m_1 m_2/r##. In other words, added to the initial potential energy, you get a total energy of 0, which means that the mass ##m_1## will not stop before reaching potential energy 0 at "infinite" or "very far away" distance.

Perhaps that will help?

The intent of the phrase "very far away" refers to the zero point of potential energy. So you can just use 0 for PE.

r1 = h
r2 = "very far away"

Sorry, forgot to define those values.

Yes, that helped! So Uf is essentially zero which then gives me the right answer.

Thank you.

## 1. What causes a rocket to burn out at a certain altitude?

A rocket burns out at a certain altitude due to the depletion of its fuel supply. As the rocket ascends, it uses up its fuel to power its engines, and once the fuel is exhausted, the rocket is no longer able to generate thrust and maintain its upward trajectory.

## 2. How does the Earth's atmosphere affect a rocket's burnout altitude?

The Earth's atmosphere plays a crucial role in a rocket's burnout altitude. As a rocket travels through the atmosphere, it experiences air resistance, which can slow down its ascent and cause it to burn out at a lower altitude. Thicker and denser atmospheres will have a greater impact on a rocket's burnout altitude.

## 3. Can a rocket's burnout altitude be predicted?

Yes, a rocket's burnout altitude can be predicted based on its initial velocity, mass, and the amount of fuel it carries. Scientists and engineers use mathematical models and simulations to estimate a rocket's burnout altitude and plan its trajectory accordingly.

## 4. How does the shape and design of a rocket affect its burnout altitude?

The shape and design of a rocket can greatly influence its burnout altitude. A streamlined and aerodynamic design can reduce air resistance and allow the rocket to reach higher altitudes before burning out. Additionally, the type and efficiency of the rocket's engines can also impact its burnout altitude.

## 5. Is the burnout altitude of a rocket the same for every launch?

No, the burnout altitude of a rocket can vary depending on several factors such as weather conditions, payload weight, and launch location. Each launch is unique, and the burnout altitude may differ based on these variables.

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