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Due to some cosmic event the earth's rotation and revolution,Sun pulls Earth toward its center i)Find the velocity v when it's distance is x from sun's center ii)Find the Time taken to reach the velocity v and The time taken to reach the sun's center.

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As the Earth is revolving around the Sun under the influence of a centripetal acceleration "a", with an angular velocity "ω" at a time period T. Let the radius of Earth's orbit be R, Ms be the mass of Sun and the mass of Earth be Me. Let G be the universal Gravitational constant.Centripetal force is supplied by the gravitational attraction force. Hence,

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Let us assume that the gravitational force of attraction on the Earth pulls it towards the Sun and the Earth starts from an initial velocity 0 at t=0. Let us assume that Earth starts from rest. (no rotational or other kinetic energy).

Force on the Earth at a distance R from the Sun = Fg = G Ms Me / R² Work done by F for moving Earth to a distance x from R = Wg =

Kinetic energy gained by Earth = Change in gravitational PE = Wg

Where x₀ = R = distance of Earth from Sun at t = 0.

========================== To derive expression for t in terms of x and v.

So time to reach the Sun is T/4√2 ======================== Time duration for Earth to reach velocity v.