A straight, smooth tunnel is dug through a uniform spherical planet of mass density . The tunnel passes through the center of the planet and is perpendicular to the planet’s axis of rotation, which is fixed in space. The planet rotates with a constant angular speed ω such that objects in the tunnel have no apparent weight. What is the required angular speed ω of the planet?

You need to be more specific.  i really don't know what a "motion map" is.you can draw a graph of an object's position, distance, displacement, speed,velocity, or acceleration versus time, or you can draw a graph of the object's x-position versus its y-position and not give any information about time at all.if the graph presents position or displacement against time,  then constant velocity is a straight line . . i.e. a line that has  constant slope.if the graph shows speed against time, then constant velocityis a horizontal piece of line.if the graph shows acceleration against time, then constant  velocity is a piece of line that lies on the x-axis . . it's value is zero.