$\"\"$

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The equation of parabolic trajectory of the falling prey is $\"\"$ .

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Differentiate with respect to $\"\"$ .

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$\"\"$

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Arc length length formula : $\"\"$ .

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$\"\"$

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Solve for $\"\"$ .

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$\"\"$

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At $\"\"$ the value of $\"\"$ is zero.

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$\"\"$

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Consider $\"\"$ .

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Let $\"\"$ .

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Differentiate with respect to $\"\"$ .

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$\"\"$

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If $\"\"$ then $\"\"$ .

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If $\"\"$ then $\"\"$ .

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Substitute corresponding values in the above equation.

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$\"\"$

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Let $\"\"$ .

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Differentiate with respect to $\"\"$ .

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$\"\"$

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Substitute corresponding values in the above equation.

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$\"\"$

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$\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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Integral reduction : $\"\"$ .

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$\"\"$

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Thus, $\"\"$ .

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Substitute equation (2) in equation (1).

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ in the above equation.

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$\"\"$

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$\"\"$

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Therefore, the distance traveled by the prey is $\"\"$ .

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$\"\"$

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The distance traveled by the prey is $\"\"$ .