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A proton moves along the $x -$ axis according to the equation $t = \sqrt{\frac{x - 50 t}{10}}$ , where $x$ is in metres and $t$ is in seconds. Calculate the average velocity of the proton during the first $3.0 s$ of its motion.

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$80 m / s$

$65 m / s$

$110 m / s$

$82 m / s$

answer is C.

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Detailed Solution

Given, $t = \sqrt{\frac{x - 50 t}{10}}$

or $x = 50 t + 10 t^{2}$

Instantaneous velocity,

$v = \frac{d x}{d t} = 50 + 20 t$

The average velocity,

$v = \frac{\int_{0}^{3} v d t}{3} = \frac{\int_{0}^{3} (50 + 20 t) d t}{3} = \frac{1}{3} {|50 t + \frac{20 t^{2}}{2}|}_{0}^{3} = 80 m / s$

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