Acceleration, Velocity and Kepplers Laws

[salsasunrise123]

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Hello, In TBR, Keplers 2nd law of orbitals says that as radius decreases, force, acceleration, and velocity increase. I get why force increase via gravity equation, but as for acceleration and velocity I am confused because I thought if you decrease radius both velocity and acceleration decreases because it takes less time to make a revolution.

Also, I am confusing myself about interpreting proportions. For example, W (ang. velocity)=V/R. Here velocity and radius are inversely proportional, but if I rearrange it to solve for velocity I get V=WR, in which case velocity and radius are directly proportional. Which is it? I'm confused.

 

[Cmdr_Shepard]

If you decrease the radius of an orbiting object, the gravitational force increases - not linearly, but by the square of the distance.

Then, according to F= ma(centrip) and F = mv^2/R, this larger Fg contributes to a larger centripetal acceleration and thus a larger tangential velocity. It takes less time to make a revolution because of the increase in velocity mostly.

As for your second paragraph, W = V/R shows that V and R are directly related (even though they are a quotient, they are on the same side of the equation - think about what you have to do to R if you increase V by 2 in order to keep W the same).