A ball approaches a batsman horizontally at speed $v_{\text{ball}}$, and the batsman swings the bat, hitting the ball back along the same path. If the speed of the bat is $v_{\text{bat}}$ when it hits the ball, then what is the recoil speed of the ball?

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**Details and Assumptions:**

- The collision is perfectly elastic.
- For all practical purposes, the speed of the bat is constant throughout the collision.

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