Angular velocity is the rate at which a point on the turntable rotates about its axis. This rate is measured as a change in the angular position divided by a change in time, Δθ/Δt. The linear velocity refers to a distance traveled per unit of time. It is sometimes referred to as the tangential velocity for an object moving in a circle.
These two velocity quantities are related. The linear velocity (v) for an object rotating in a circle of radius R is related to the angular velocity (ω) and the radius of the circle (R). The equation relating these quantities is
v = ω*R.
Since this question pertains to angular velocity, it is often useful to re-arrange the equation to the form of
ω = v/R.
This form of the equation leads to the claim that the angular velocity is directly proportional to the linear velocity and inversely proportional to the radius. In this question, one of the buckets has twice the linear velocity of the other bucket. This would lead to twice the angular velocity. The same bucket has one-half the radius. Since angular velocity and radius are inversely proportional, one-half the radius would also lead to twice the angular velocity. When combined, these two factors and their effects would lead to one object having four times the angular velocity of the other object.
One way to conceptualize this is to think of angular velocity as the rate at which the angular position changes (Δθ/Δt). An object that is traveling twice as fast will change the angular position at twice the rate. And an object that is traveling around a circle with one-half the radius (and one-half the circumference) only has to travel half as far to produce the same angular position change. So when both factors are put together - twice the linear speed and one-half the radius - one bucket would complete an entire 360-degree revolution (2•π radians) in one-fourth the time. It thus has four times the angular velocity since it does a 360-degree angular position change in one-fourth the time.