MHT CET202210 Aug 2022Evening ShiftPhysicsGravitationActual
The mass and radius of the earth and moon are M₁, R₁ and M₂, R₂ respectively. Their centres are at a distance d apart. The minimum speed with which a body of mass m should be projected from a distance ( 2 d 3 ) from the centre of M₁ so as to escape to is
Options
- A[ 3 G (M₁+2 M₂ ) d ] 1 2
- B[ 3 G (M₁-M₂ ) 2 d ] 1 2
- C[ 6 G (M₁-M₂ ) 2 d ]^ 1 2
- D[ 6 G (M₁+M₂ ) d ]^ 1 2
Correct answer
A. [ 3 G (M₁+2 M₂ ) d ] 1 2
Step-by-step solution
If the mass m is at a distance 2 3 d from the centre of M₁ then it is located at a distance d 3 from the centre of M₂ . To calculate the kinetic energy required to make the body escape Earth and moon system: the change in gravitational potential energy between final and initial location should be equal to the kinetic energy given to the mass m . aligned & 1 2 m v_e^2= [ G M₁ m ( 2 d 3 ) + G M₂ m ( d 3 ) ] & 1 2 m v_e^2= 3 G (M₁+2 M₂ ) m 2 d aligned Therefore, escape velocity v_e= 3 G (M₁+2 M₂ ) d