An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 43.0 km . To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 23.0 m/s .If nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?

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Question:

An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 43.0 km . To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 23.0 m/s .If nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?

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Answer: v₂ = 5962 kmthe spacecraft  will crash into the lunar surface at a speed of 5962 km if nothing is done to correct its orbitsExplanation:Given that;Lunar surface is in an altitude h = 43.0 km =  43 × 10³ mwe know; Radius of moon R₁ = 1.74 × 10⁶, mass of moon = 7.35 × 10²²speed of the space craft when it crashes into the lunar surface , vdecreasing speed of the space craft = 23 m/sNow since the space craft travels in a circular orbit, we use centrifugal expression Fe = mv²/rbut the forces is due to gravitational forces between space craft and lunar surface Fg = GMn/r²HERE r = Rm + hwe substituter = 1.74 × 10⁶ m + 43 × 10³ m= 1.783 × 10⁶ mOn equating these, we haveG is gravitational force ( 6.673 × 10⁻¹¹ Nm²/kg²)v²/r = GM/r²v = √ ( GM/r)v = √ ( 6.673 × 10⁻¹¹ Nm²/kg² × 7.35 × 10²² / 1.783 × 10⁶ )v = √ (2750787.9978)v = 1658.55 m/sNow since speed is decreasing by 23 m/sthe speed of the space craft into the lunar face is,v₁ = 1658.55 m/s - 23 m/sv₁ = 1635.55 m/sNow applying conversation of energy, we say1/2mv₂² = 1/2mv₁² + GMem (1/Rm - 1/r)v₂ =  √ [ v₁² + GMe (1/Rm - 1/r)]v₂ =   √ [ 1635.55²  + ( 6.673 × 10⁻¹¹ Nm²/kg² × 7.35 × 10²²) (1/ 1.74 × 10⁶ - 1 / 1.783 × 10⁶)]v₂ =  √ (2675023.8025 + 67979.24)v₂ = √(2743003.046)v₂ = 1656.2 m/snow convertv₂ = 1656.2 × 1km/1000m × 3600s/1hrsv₂ = 5962 kmTherefore the spacecraft  will crash into the lunar surface at a speed of 5962 km if nothing is done to correct its orbits

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