Motion in Straight Line

Question

Two stars each of one solar mass (= 2 x 10³⁰ kg) are approaching each other for a head-on collision. When they are a distance of 10⁹km, 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. (G = 6.67 x 10^-11Nm²/kg²)

Solution

Let $stack straight v subscript 1 with rightwards arrow on top space and space stack straight v subscript 2 with rightwards arrow on top$ be the velocities of two stars when they collide.
According to the law of conservation of momentum,
$straight M stack straight v subscript 1 with rightwards arrow on top plus straight M stack straight v subscript 2 with rightwards arrow on top equals 0$
$rightwards double arrow$ $stack straight v subscript 1 with rightwards arrow on top space equals space minus stack straight v subscript 2 with rightwards arrow on top$
$rightwards double arrow$ $space straight v subscript 1 equals straight v subscript 2 equals straight v$
According to the law of conservation of energy,
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We have
Mass of the star, $straight M equals 2 cross times 10 to the power of 30 kg$
Distance between the stars, $straight r subscript 1 equals 10 to the power of 9 km equals 10 to the power of 12 straight m$
Radius of star, $straight r equals 10 to the power of 4 km equals 10 to the power of 7 straight m$
The distance between two stars when they collide is,
2r = 2 x 10⁷m
∴ Speed with which the stars collide is given by,

$straight v equals square root of GM open square brackets 1 over straight r subscript 2 minus 1 over straight r subscript 1 close square brackets end root$
$equals square root of 6.67 cross times 10 to the power of negative 11 end exponent cross times 2 cross times 10 to the power of 30 cross times open square brackets fraction numerator 1 over denominator 2 cross times 10 to the power of 7 end fraction minus 1 over 10 to the power of 12 close square brackets end root$
$equals 2.58 cross times 10 to the power of 6 space straight m divided by straight s$

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Motion In Straight Line

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