Can You Calculate the Trajectory of an Object Orbiting Earth at High Speed?

[chui]

An object starts at a velocity of 12,000meter/sec,
(the direction tangent to an orbit around earth)
from a distance of 40,000km from Earth's center.

How can I calculate its track?
thnks.

 

[Severian596]

Isn't the object's mass necessary for this type of problem? I believe that at speeds like this, a "massless" point particle would escape Earth's gravity and travel in a path that is almost straight (but slightly curved due it its very small relativistic mass). The first thing you should do is plot the force vectors, and without mass this isn't possible.

The resultant vector will be sufficient to keep the object in orbit (at equilibrium), or it will signify an escape, or it will signify a descent towards earth. You'll need to plot its position as a function of time to plot its path. Make the center of the Earth the origin. Make the mass at (40, 0), and make the x-axis measured in kilometers. Initial vector heads in the +y direction. What's its magnitude? What's the magnitude of the force toward the origin? How does the position of the point change after 1 second, based on its acceleration towards the origin?

 

[lightgrav]

Do you want to *simulate* this path
(location as function of time, step-by-step)
Or do you want to describe the orbit*
(apogee radius and speed, perigee r & v)?

VPython does a nice job with the momentum vector,
step-by-step changing it according to F(r_vec)

Can you find its KE, total E, and angular momentum?

 

[HallsofIvy]

No, the object's mass is not necessary (assuming it does have mass!) since, as with all gravity problems, the m cancels in $F= -\frac{GMm}{r^2}= ma$.

One way to do this would be to solve the differential equations for the motion- preferably in polar coordinates but they are non-linear. A good book on oribital motion should have an example of this.

Probably better is to assume the orbit is an ellipse, write down the general equation of an ellipse (again, I would use polar coordinates) and use conservation of energy to determine the coefficients.

 

[chui]

One way to do this would be to solve the differential equations for the motion- preferably in polar coordinates but they are non-linear.

ok, tell me if what I am doing here is fine becous I think it isnt:
for the R axis (Polar cordinatsr) it should be:
v^2/r - MG/r^2 = F(r)
now I can integrate this function (dr) from 40,000,000 to wherever I want the final distance from Earth to be. and get the acceleration on this axis.

for the angle Theta I can 360*v*t/2(pai)r
but then I don't what to do with the variable r.

Probably better is to assume the orbit is an ellipse

well, does it? (I thoght it will be some kind of an going-out Spirall)