by apodman » Sat Jul 19, 2008 6:34 pm
BMAONE23 wrote:orin stepanek wrote:... probably have to collide with something big.
... Whether it would stop us or merely slow us, and how quick the affect would be, I couldn't guess
(elsewhere) henk21cm wrote:follow my motto: "start simple, complicate later"
I picture the Earth running into an object of its own mass fast enough to produce an inelastic collision. If this object is also rotating with the same angular momentum as the Earth in the
same direction (the opposite of two meshed gears that rotate in
opposite directions), it will exactly stop the Earth's rotation. The effect would be immediate.
Now we can vary the speed of impact, angle of impact, and the impact's offset from center to introduce the conditions of a grazing impact. Since the gravity of the two bodies will pull them together at a speed at least equal to the escape velocity of the more massive body (I think), not all speeds and angles are permissible. Separate problems may need to be set up for inelastic and partially elastic collisions. The permissible values still have three variables to contend with (speed, angle, offset), so some 3D minimax calculus (gradient, etc.) will likely be needed to find the impact vector that requires the smallest object to stop the Earth's rotation. I'm not sufficiently curious to solve the problem myself, being more of "a solution exists" kind of guy.
I might only introduce the fourth variable (rate of the object's rotation) after solving the problem above.
In my quick reading of henk21cm's equations above (unless I missed something), the objects impacting the Earth are not rotating themselves. If the nature of the collision allows the conversion of rotational inertia into momentum, then the smaller the impacting body and the less complete the conversion, the faster it needs to be rotating to produce the same result (or a larger body with more complete conversion rotating more slowly). In other words (in theory), the Earth could run into a vast immovable object and stop with a slight grazing impact, or run into a small rock spinning very very fast and halt its rotation completely.
A non-spinning object could impact the Earth in such a manner that it is sent away spinning, thereby carrying off some of the momentum from the collision as rotational inertia and complicating our problem just a little bit more.
Note for those not familiar with "elastic" and "inelastic". If two flying gobs of peanut butter hit each other and stick together, the collision is inelastic. If two hard rubber balls (or subatomic particles) bounce off each other, together retaining a sum of 100% of their incoming kinetic energy, the collision is elastic. If you drop a billiard ball on a resilient floor, you can tell that the collision is not perfectly elastic because the ball does not bounce back to its original height, and because you can hear that some of the energy of the collision has been converted into sound. An "elastic" collision conserves
kinetic energy - see
http://hyperphysics.phy-astr.gsu.edu/Hbase/elacol2.html and
http://hyperphysics.phy-astr.gsu.edu/Hb ... on.html#c1 . A "collision" of galaxies in not a true collision at all in this sense (and therefore the concept of elasticity doesn't apply), since their masses do not interact as two objects colliding.
[quote="BMAONE23"][quote="orin stepanek"]... probably have to collide with something big.[/quote]
... Whether it would stop us or merely slow us, and how quick the affect would be, I couldn't guess[/quote]
[quote="(elsewhere) henk21cm"]follow my motto: "start simple, complicate later"[/quote]
I picture the Earth running into an object of its own mass fast enough to produce an inelastic collision. If this object is also rotating with the same angular momentum as the Earth in the [i]same[/i] direction (the opposite of two meshed gears that rotate in [i]opposite[/i] directions), it will exactly stop the Earth's rotation. The effect would be immediate.
Now we can vary the speed of impact, angle of impact, and the impact's offset from center to introduce the conditions of a grazing impact. Since the gravity of the two bodies will pull them together at a speed at least equal to the escape velocity of the more massive body (I think), not all speeds and angles are permissible. Separate problems may need to be set up for inelastic and partially elastic collisions. The permissible values still have three variables to contend with (speed, angle, offset), so some 3D minimax calculus (gradient, etc.) will likely be needed to find the impact vector that requires the smallest object to stop the Earth's rotation. I'm not sufficiently curious to solve the problem myself, being more of "a solution exists" kind of guy.
I might only introduce the fourth variable (rate of the object's rotation) after solving the problem above.
In my quick reading of henk21cm's equations above (unless I missed something), the objects impacting the Earth are not rotating themselves. If the nature of the collision allows the conversion of rotational inertia into momentum, then the smaller the impacting body and the less complete the conversion, the faster it needs to be rotating to produce the same result (or a larger body with more complete conversion rotating more slowly). In other words (in theory), the Earth could run into a vast immovable object and stop with a slight grazing impact, or run into a small rock spinning very very fast and halt its rotation completely.
A non-spinning object could impact the Earth in such a manner that it is sent away spinning, thereby carrying off some of the momentum from the collision as rotational inertia and complicating our problem just a little bit more.
Note for those not familiar with "elastic" and "inelastic". If two flying gobs of peanut butter hit each other and stick together, the collision is inelastic. If two hard rubber balls (or subatomic particles) bounce off each other, together retaining a sum of 100% of their incoming kinetic energy, the collision is elastic. If you drop a billiard ball on a resilient floor, you can tell that the collision is not perfectly elastic because the ball does not bounce back to its original height, and because you can hear that some of the energy of the collision has been converted into sound. An "elastic" collision conserves [i]kinetic[/i] energy - see http://hyperphysics.phy-astr.gsu.edu/Hbase/elacol2.html and http://hyperphysics.phy-astr.gsu.edu/Hbase/colcon.html#c1 . A "collision" of galaxies in not a true collision at all in this sense (and therefore the concept of elasticity doesn't apply), since their masses do not interact as two objects colliding.