If a planet was suddenly stopped in its orbit supposed to be circular, show that it would fall onto the sun in a time equal to (sqrt2 //8)times the period of planet.s revolution.

If a planet was suddenly stopped in its orbit supposed to be circular,

1.	If a planet was suddenly stopped in its orbit supposed to be circular, show that it would fall onto the sun in a time equal to (sqrt2 //8)times the period of planet.s revolution.
Answer» Solution :If M is mass of the SUN and r is radius of the planet.s orbit, its orbital SPEED is `V_0 = sqrt((GM)/(r ))` Orbital time period `T = (2pi r)/(v_0) " or " T^2 = (4pi^2 r^3)/(GM)` After stopping the planet, if it has velocity V, when it is at a distance x from the sun, from conservation of MECHANICAL energy, ` 1/2 mv^2 - (GMm)/(x) = - (GMm)/(r ) IE, - ((dx)/(dt))^2 = (2GM)/(r ) [ (r - x)/(x)]` `(-dx)/(dt) = sqrt((2GM)/(r )) sqrt((r-x)/(x)) ,int_0^t dt = - sqrt((r)/(2GM)) int_r^0 ( sqrt((x)/(r - x)) )dx ` After solving , we get `t = (sqrt2)/(8) T`