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Thread: Physics problem - Geosynchronous Potato Cannon

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    Physics problem - Geosynchronous Potato Cannon

    All fun is to be found in physics. Here is an interesting physics problem relating to geosynchronous orbits. Put a potato cannon in geosynchronous orbit and aim it toward the center of the earth (in other words, perfectly vertically). What kind of path would a launched potato follow in relation to the platform? In relation to the earth's surface? For the sake of this thought experiment, you can eliminate the earth's atmosphere and treat the entire translation from launch to impact (if any) as occurring in a vacuum.

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    Quote Originally Posted by thorkil2 View Post
    All fun is to be found in physics. Here is an interesting physics problem relating to geosynchronous orbits. Put a potato cannon in geosynchronous orbit and aim it toward the center of the earth (in other words, perfectly vertically). What kind of path would a launched potato follow in relation to the platform? In relation to the earth's surface? For the sake of this thought experiment, you can eliminate the earth's atmosphere and treat the entire translation from launch to impact (if any) as occurring in a vacuum.
    We should ignore the roughness of the earth gravity field as well, treat it as spherically symmetric.

    The velocity of a potato cannon is tiny compared to orbital speed, even at geosynch orbit, and such a small perpendicular deflection only slightly modifies the potato's path, into a slightly eccentric ellipse, that will eventually return to that place in the orbit, ignoring other perturbations. The potato cannon probably won't still be there.

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    Might the potato cannon return to that same spot from its higher orbit, though, when the potato comes back? Launch site would be perigee, apogee for potato.
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    My understanding is: if we designate the firing point in the orbit as zero degrees, the potato will initially lose altitude until it reaches the 90 point, where it will begin gaining altitude. It will cross the potato gun's orbit* at 180, gain altitude until 270, then lose altitude and cross again at 0.

    *...or initial orbit, if we choose not to ignore conservation of momentum.
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    Quote Originally Posted by grapes View Post
    We should ignore the roughness of the earth gravity field as well, treat it as spherically symmetric.

    The velocity of a potato cannon is tiny compared to orbital speed, even at geosynch orbit, and such a small perpendicular deflection only slightly modifies the potato's path, into a slightly eccentric ellipse, that will eventually return to that place in the orbit, ignoring other perturbations. The potato cannon probably won't still be there.
    We could make it more interesting by launching the potato at something close to the orbital speed, say just over 3 km/s. The question involves only the potato. We can discard the launcher to the cloud of other anonymous space junk.

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    You never cancel the orbit speed with a vertical radial launch so you have a coriolis situation, at lower altitude the potato is going too fast to stay on its radial path, so it also too fast tangentially to be in a lower orbit so it goes eccentric as said above. To follow a radial path it would have to be continuously decelerated tangentially.
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    Exactly, which is what makes the problem interesting. It would impossible to aim a projectile vertically at a point directly below the launch point and hit the target point (atmospheric interference not included in this consideration), without continuous course adjustment. Essentially, there is no such thing as straight down.

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    Quote Originally Posted by thorkil2 View Post
    We could make it more interesting by launching the potato at something close to the orbital speed, say just over 3 km/s. The question involves only the potato. We can discard the launcher to the cloud of other anonymous space junk.
    Escape velocity is root two times circular velocity. If you launch your potato at right angles to the orbital velocity vector, with a velocity of the same magnitude as the orbital velocity, then you've given it escape velocity, directed at 45 degrees pitch down. It'll follow a parabola, pass through perigee, and then escape. Lower launch velocities produce ellipses, with perigee inside your circular orbit, and apogee outside - the lower the launch velocity, the lower the eccentricity of the ellipse. Higher launch velocities will give you a hyperbola. There will be a high value of launch velocity at which perigee grazes the Earth, and launch velocities higher than that will result in an Earth impact.

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