Solve Satellite Problem: Find Speed at Maximum Height

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In summary, the conversation discusses the calculation of the speed of a small package that is fired off Earth's surface and reaches a maximum height of h above the surface, which is equal to Earth's radius. Different equations and concepts such as energy conversion, parabolic motion, and angular momentum are used to calculate this speed, but the results do not reconcile due to the assumption of constant acceleration of gravity in some equations.
  • #1
yasar1967
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A small package is fired off Earth's surface with a speed V at a 45o angle. It reaches maximum height of h above surface at h=6370km, a value equal to Earth's radius itself.
What is the speed when it reaches this height? Ignore any effects that might come from Earth's rotation.




Energy converstion: 1/2 mv^2 - GmM/R1 = 1/2mV^2 - 1/2 GmM/R2
Parabolic Motion: hmax= vo^2(sin^2 ø) /2g
Angular momentum conversition :L=rmv etc. etc.



3. I first calculated the Vo speed from hmax= vo^2(sin^2 ø) /2g equation
Then I put the value into:
1/2 mv^2 - GmM/R1 = 1/2mV^2
- 1/2 GmM/R2

noting that R2=hmax=2 x Rearth

Then I calculated and found the SAME result from parabolic motion equations:
first
time= x/V0 cosø ... and then finding out Vx and Vy thus resulting to V speed.

Yet,

The book starts with angular momentum conversion, stating that :
m(Vcosø)R = m Vfinal 2R and then using Vescape speed formula and energy conversition it comes out with completely different result.

Where did I go wrong? or how come results do not reconcile?
[/b]
 
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  • #2
hmax = vo^2(sin^2 ø) /2g is only valid if the acceleration of gravity is constant.

You can use the angular momentum equation to get the velocity at the apogee of the orbit as a function of the velocity at the surface. You can then subtitute that in the energy equation
 
  • #3

There are a few things to consider in this problem. First, it is important to clarify what is meant by "maximum height." Is this referring to the highest point in the satellite's trajectory, or the point where its speed is at its maximum? This distinction can lead to different results.

Assuming that "maximum height" refers to the highest point in the trajectory, your approach using energy conversion and parabolic motion equations seems correct. However, it is important to make sure that all the variables and equations used are consistent with each other.

For example, in the energy conversion equation, you have used the radius of the Earth (R1) and the radius of the satellite's orbit (R2). However, in the parabolic motion equation, you have used the height (hmax) instead of the radius of the orbit. This could lead to a discrepancy in the results.

Additionally, in the angular momentum conversion approach, it is important to make sure that the angular momentum is conserved throughout the entire trajectory, not just at the highest point. This may require using more advanced equations and considering the changing velocity and position of the satellite.

In summary, it is important to use consistent equations and variables, as well as consider the entire trajectory of the satellite, in order to get accurate results for the speed at maximum height. It may also be helpful to double check your calculations and equations to identify any errors.
 

Related to Solve Satellite Problem: Find Speed at Maximum Height

What is the "Solve Satellite Problem"?

The "Solve Satellite Problem" is a mathematical problem that involves determining the speed of a satellite at its maximum height or apex. It is commonly used in physics and engineering to calculate the trajectory and orbit of satellites.

How is the speed at maximum height calculated?

The speed at maximum height can be calculated using the equation v = √(GM/R), where v is the speed, G is the gravitational constant, M is the mass of the planet, and R is the distance between the satellite and the center of the planet. This equation is derived from the conservation of energy and angular momentum principles.

What factors affect the speed at maximum height?

The speed at maximum height is affected by the mass of the planet the satellite is orbiting, the distance between the satellite and the planet, and the gravitational constant. Other factors that can affect the speed include atmospheric drag, solar radiation, and the shape and orientation of the satellite's orbit.

What are some real-world applications of solving the satellite problem?

Solving the satellite problem has many real-world applications, including predicting the trajectory and orbit of satellites, launching and controlling spacecrafts, and designing satellite missions for communication, navigation, weather forecasting, and other purposes. It is also used in the aerospace industry for space exploration and satellite-based research.

Are there any limitations to solving the satellite problem?

Yes, there are some limitations to solving the satellite problem. The equation used to calculate the speed at maximum height assumes a perfect circular orbit and does not take into account external forces such as atmospheric drag. Also, the mass of the planet and the distance to the satellite may vary, which can affect the accuracy of the calculation. In addition, the satellite's orbit can be influenced by other objects in space, making it difficult to predict its exact speed at maximum height.

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