We consider a falling particle whose mass is given by m. we assume that the forces acting on the particle are both the air resistance, which is proportional to the velocity of the particle, and the gravitation. the position of the particle at time t is given by x(t) and the direction of the gravitation is negative x direction. the equation of motion of the particle is given by d2r(t) dr(t) (1) mg a dt m dt2 11 where a is a positive constant, and g is the magnitude of gravitational acceleration we assume that the initial position of the particle is given by x(0) = 0 and the initial velocity of the particle is 0. answer the following questions. as the differential equation of the velocity v(t) of the particle (1) rewrite eq. (1) (2) when t oo, the acceleration becomes 0, and the velocity reaches the terminal velocity v(oo). express the terminal velocity using m, g and a. (3) we introduce a new variable u(t) v(t) - v(o0). rewrite eq. (1) as the differential equation of u(t) and express u(0) using m, g and a (4) express ut) as a function of t. (5) express v(t) as a function of t (6) select the correct statement from the following choices: (a) the mechanical energy of the particle increases with increasing time (b) the mechanical energy of the particle decreases with increasing time. (c) the mechanical energy does not change during the fall