Solve Satellite Problem: Find Speed at Maximum Height

[yasar1967]

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]

 

[kamerling]

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

 

[Moetasim]

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.