Two stars each of one solar mass (= 2×1030 kg) are approaching each other for a head-on collision. When they are a distance 109 km, their speeds are negligible. What is the speed with which they collide? The radius of each star is 104 km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
The mass of a star is, M=2×1030 kg The radius of each star is, R=104 km=107 m Distance between two stars is,r=109 km=1012 m As the stars are separated by a distance, initially the stars will have gravitational potential energy. Total energy of the system of two stars is given by: E=−rGMM ......(1) Now as the stars move towards each other they gain the kinetic energy. Let velocity of the stars when they are about to collide be v and at collision the distance between the centers of the stars =2R The total kinetic energy of the stars system Ek=21Mv2+21Mv2=Mv2 and the total potential energy of the two stars system close to each other is: Ep=−2RGMM Thus, the total energy of the two stars just before collision Et=Mv2−2RGMM ........(2) Since the energy remains conserved, the initial and final energy will be same. Mv2−2RGMM=−rGMM v2=2RGM−rGM =6.67×10−11×2×1030(2×1071−10121) ≈6.67×1012 ∴v=2.58×106 m/s
