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Q.

A particle is moving on a circular path of radius 0.5m with a tangential deceleration that is always half of the centripetal acceleration. The initial velocity is 20 m/s. If speed of the particle at $t = 5$ is $\frac{4 x}{101} m / s,$ then find the value of x .

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answer is 5.

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Detailed Solution

As, $- \frac{d v}{d t} = \frac{v^{2}}{2 R} = \frac{v^{2}}{2 \times 0.5}$ or $- \frac{d v}{d t} = v^{2}$ or $- \int_{20}^{v} \frac{d v}{v^{2}} = \int_{0}^{5} d t$ or ${[\frac{1}{v}]}_{20}^{v} = 5$ or $\frac{1}{v} - \frac{1}{20} = 5$ or $\frac{1}{v} = \frac{1}{20} + 5 = \frac{101}{20}$

$∴ v = \frac{20}{101} = \frac{4 \times 5}{101}$

According to the equation $\frac{4 \times 5}{101} = \frac{4 x}{101}$

$x = 5$

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