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When the ball travels in a circular path there is a constant acceleration acting on it that maintains the circular path. The tension in the string is what leads to this acceleration. In the absence of this, the ball would move in a linear path.
The linear velocity of a body traveling in a circular path at any time is given by v = 2*pi*R/T where T is the period for one rotation and R is the radius of the path it is rotating in.
When the ball rotates at 46 rpm, the time period for one rotation is 60/46 s. The radius of the path R = 10 m.
Susbtituting these values in v = 2*pi*R/T
=> v = 2*pi*10*46/60 m/s
=> v = 48.17 m/s
If the string is cut the ball travels with a linear speed equal to 48.17 m/s.
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