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I'm trying to make a game where the character can swing on a rope.

I want to know how the velocity of player is calculated when it stops circular motion. Any help would be appreciated.

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This picture exemplifies it well. In the given example the rope pulls the robot with the blue vector (a force in the direction of the rope). Once you let go of the rope, of there wass no gravity the robot would move with equal speed in a straight line tangential to the point he stopped the circular motion. If there's gravity, the only acceleration on the robot is the green vector pointing down.

In realistic accelerated movement, it's often better to control only the acceleration of the robot, and determine velocity as a by-product of acceleration, using kinematic equations.

enter image description here

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Circular motion already has velocity; what makes it circular is the continued application of a rotating acceleration of constant magnitude. So to let the object leave the circle at the proper velocity, just stop applying the acceleration.

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    \$\begingroup\$ The acceleration is only constant (actually it is zero) from the rotating frame of reference centered on the circle and rotating at the same angular velocity. From a typical fixed, non-rotating frame of reference only the magnitude of the acceleration is constant; it's direction is always from the object to the center of the orbit, and thus is always changing. \$\endgroup\$ – bcrist Nov 9 '14 at 19:44
  • \$\begingroup\$ That was of course what I meant I will edit to clarify. :) \$\endgroup\$ – Almo Nov 10 '14 at 1:01

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