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Homework answers / question archive / Show that the areal velocity is constant for a particle moving under the influence of an attractive force given by F(r) = -kr
Show that the areal velocity is constant for a particle moving under the influence of an attractive force given by F(r) = -kr. Calculate the time averages of the kinetic and potential energies and compare the results of the viral theorem.
Areal velocity w = (r * V)/2
Where * denotes the vector product, r is the position of the particle and V its velocity.
dw/dt = (dr/dt) * V/2 + (r/2) * (dV/dt)
Now dr/dt = V
Further dV/dt = acceleration = Force/Mass
Force F = -k r (i.e., it is along the radius vector)
Therefore dV/dt = - (k/m) r
Plugging it back in we have
dw/dt = V * V/2 - (k/2m) r * r
But V*V = r*r = 0
Therefore dw/dt = 0
or w is a constant in time.
