Question

Two spherical asteroids have the same radius R. Asteroid 1 hasmass M and asteroid 2 has mas 2M. The two asteroids are releasedfrom rest with distance 10R between their centers. What is thespeed of each asteroid just before they collide?

Answer 1

Answer:

 $v_1 =\sqrt{\dfrac{16GM}{15R}}$

 $v_2 =\sqrt{\dfrac{4GM}{15R}}$

Explanation:

given,

mass of asteroid 1 = M

mass of asteroid 2 = 2M

radius of two asteroid = R

Distance between the asteroid = 10 R

Speed of the asteroid before collision = ?

using conservation of momentum

M u + 2M u' = M v₁ + 2 M v₂

initial speed of asteroid is equal to zero

0 = v₁ + 2 v₂

v₁ = -2 v₂

using conservation of momentum

initial potential energy is converted into potential energy and the kinetic energy of both the asteroids.

 $\dfrac{GM(2M)}{10R}=\dfrac{GM(2M)}{2R}+\dfrac{1}{2}Mv_1^2 + \dfrac{1}{2}(2M)v_2^2$

 $\dfrac{GM(2M)}{10R}-\dfrac{GM(2M)}{2R}=\dfrac{1}{2}M(-2v_2)^2 + \dfrac{1}{2}(2M)v_2^2$

 $6v_2^2 = \dfrac{8GM}{5R}$

 $v_2 =\sqrt{\dfrac{4GM}{15R}}$

now,

 $v_1 =-2\sqrt{\dfrac{4GM}{15R}}$

 $v_1 =\sqrt{\dfrac{16GM}{15R}}$

hence, the velocity of asteroid are

 $v_1 =\sqrt{\dfrac{16GM}{15R}}$

 $v_2 =\sqrt{\dfrac{4GM}{15R}}$