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?

Answer 1

Mass of a particle depends on the relativistic speed v with which it moves as $m=\frac{m_o}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}$.
The gravitational force of attraction of Earth is given by

$$\begin{gathered} F=\frac{GMm}{r^2} \\ \Rightarrow F=\frac{GM{m}_o}{r^2\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}} \end{gathered}$$

Note: This is the only reason responsible for gravitational lengthening in which a photon travelling at a speed c experiences gravitational force.