If an object moves in uniform circular motion in a circle of radius r = 1.0 meter, and the object takes 4.0 seconds to complete ten revolutions, calculate the magnitude of the velocity around the circle. (note: remember, 10 revolutions is a counting number and not a measurement.)

Let's supose that your object is the orange rectangle rotating around the blue circle (see picture attached)

r = 1.0 [m] is the radius of the circle (it's R in your problem).
ω [rad/s] - is the angular velocity of the object, it is measured in radians per second. We will compute it from the data you have
v [m/s] - is the tangential/linear speed, it is measured in meters per second

We know that the rotational speed is 10 revolutions per 4 seconds.

10/4 rev/s = 2.5 rev/s

We convert rev/s in rad/s and we get:
2.5 rev/s = 15.708 rad/s

So we found our angular speed: ω = 15.708 rad/s

The relationship between v and ω is:

v = ω · r = 15.708 · 1 = 15.708 m/s

  the magnitude of the velocity around the circle is 15.708 m/s

Radius  =  1.0 meter
Diameter = 2.0 meters
Circumference  =  2 π meters
10 revolutions = 20 π meters

Speed = (10 revs) / (4 sec) 

            =  (20 π meters) / (4 seconds)

             =       5 π  m/s

            =   15.708 m/s       (rounded) 

Is a  vector quantity  that refers to "the rate at which an object changes its position." imagine a person moving rapidly - one step forward and one step back - always returning to the original starting position. while this might result in a frenzy of activity, it would result in a zero velocity. because the person always returns to the original position, the motion would never result in a change in position. since velocity is defined as the rate at which the position changes, this motion results in zero velocity. if a person in motion wishes to maximize their velocity, then that person must make every effort to maximize the amount that they are displaced from their original position. every step must go into moving that person further from where he or she started. for certain, the person should never change directions and begin to return to the starting position.