Two uniform spheres x and y roll without slipping

Space-time metric ds2= -dt2+at2dx2+bt2dy2+ct2dz2 (a)how does the area of a rectangle with vertices at ( x=0, y=0, z=0 ), (1,0,0 ), ( 0,1,0 ) and (1,1,0 ) changes with time? (b) if a photon moving in the x direction starts from the origin at t=1, when does it reach x=l? (ds=0 for the trajectory of a photon.) (c) the proper time for a particle is defined by cd=ds. what is the proper time for a particle to move along the trajectory ( x(t)=t, y(t)=0, z(t)=0 ) from t=0 to 1 ? what is the condition on the parameter a so that the velocity of the particle never exceeds the speed of light ?

In the sport of parasailing, a person is attached to a rope being pulled by a boat while hanging from a parachute-like sail. a rider is towed at a constant speed by a rope that is at an angle of 15 ∘ from horizontal. the tension in the rope is 1900 n. the force of the sail on the rider is 30∘ from horizontal. what is the weight of the rider? express your answer with the appropriate units.

The speed of light can be used to measure the