A car is moving on a circular path of radius 600 m such that the magnitudes of the tangential acceleration and centripetal acceleration are equal .The time taken by the car to complete first quarter of revolution ,if it is moving with an initial speed of $54 k m / h r$ is $t (1 - e^{- π / 2}) s$ . The value of t is ____

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answer is 40.

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Detailed Solution

$\begin{array}{l} as a_{r} = a_{t} \Rightarrow {Rω}^{2} = Rα \\ \Rightarrow α = \frac{dω}{dt} = ω^{2} \Rightarrow \frac{dω}{ω^{2}} = dt \\ \Rightarrow \int_{ω_{0}}^{ω} \frac{dω}{ω^{2}} = t, where ω_{0} = \frac{v_{0}}{R} = \frac{15}{600} = \frac{1}{40} rad / \sec \end{array}$

$\begin{array}{l} \Rightarrow \frac{1}{ω_{0}} - \frac{1}{ω} = t \Rightarrow ω = \frac{1}{40 - t} = \frac{dθ}{dt} \\ \Rightarrow \int_{0}^{π / 2} dθ = \int_{0}^{t} \frac{dt}{40 - t} \Rightarrow \frac{π}{2} = [- \ln (40 - t)]_{0}^{t} \\ \Rightarrow \frac{π}{2} = \ln \frac{40}{40 - t} \Rightarrow 1 - \frac{t}{40} = e^{- π / 2} \Rightarrow t = 40 (1 - e^{- π / 2}) \end{array}$