10. If the radial velocity of a particle is proportional to its transverse velocity then the
equation of the path of the particle in polar coordinate is
a) r = Aeke, A= arbitrary constant, k=constant of proportionality
b)1= Ae-ke, A= arbitrary constant, k= constant of proportionality
c)r= Aekle, A= arbitrary constant, k=constant of proportionality
d) r= Aek, A= arbitrary constant, k= constant of proportionality
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The radial and transverse components which they are moving rapidly with an outward direction.
Where the radial velocity is an object for the path from a fixed point through a straight line.
And the transverse velocity which will refers to an object to a path with an angle θ to the origin path from a fixed point.
radical velocities ∝ transverse velocities
dr/dt ∝ r dθ/dt
dr/dt ∝ λr dθ/dt
Where λ is constant
dr/r = λ dθ
By integrating we get
log r = λθ + log a where log a is constant
Where
log r - log a = λθ
log r/a = λθ
r/a = e^{λθ}
r = ae^{λθ}
Hence it is an equiangular spiral.
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