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Question
Mass of a particle depends on its speed. Does the attraction of the earth on the particle also depend on the particle's speed?
Solution
Mass of a particle depends on the relativistic speed v with which it moves as
\[m = \frac{m_o}{\sqrt{1 - \frac{v^2}{c^2}}}\]
The gravitational force of attraction of Earth is given by
\[F = \frac{GMm}{r^2}\]
\[ \Rightarrow F = \frac{GM m_o}{r^2 \sqrt{1 - \frac{v^2}{c^2}}}\]
Notes
This is the only reason responsible for gravitational lengthening in which a photon travelling at a speed c experiences gravitational force.
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