Moving body approaching Jupiter

MOVING BODY APPROACHING JUPITER

INTRODUCTION

Jupiter is the fifth planet in our Solar System. Consider a body of mass m moving with an initial speed of 10 km/s approaching Jupiter at a distance of 5R from its surface. As the body enters Jupiter’s gravitational field, it begins accelerating. This type of acceleration is continuous acceleration as it's continuously being accelerated due to Jupiter’s gravity.

ASSUMPTION 

Jupiter has no atmosphere.

 

CALCULATION  

The final velocity of the moving body when it touches Jupiter’s surface is,

v² = u² + 2GM [(1/R) – (1/R+h)] 

Where, G = 6.67*10^-11 Nm²/kg² [Universal Gravitation Constant]

R – Radius of Jupiter [R = 69911000m]

M - Mass of Jupiter, M = 1.898*10²⁷ kg

h - Height above the surface of Jupiter, h = 5R

u - Initial velocity of the mass, u = 10km/s

v - Final velocity of the mass

v² = u² + 2GM [(1/R) – (1/R+5R)]

v² = u² + 2GM [(1/R) – (1/6R)]

v² = u² + 2GM [5R/6R²]

v² = u² + GM [5/3R] 

Substituting all values we get,

v² = 10⁸ + [3018027683]

v² = 3,118,027,683

v    = 55,839.3 m/s

CONCLUSION 

Thus the final velocity of the body on the surface of Jupiter is 55,839.3 m/s if it's moving with an initial velocity of 10km/s. The change [increase] in velocity is,

 Δ = v – u

     = 55839.3 – 10000

     = 45839.3 m/s

The percentage increase in velocity is,

∆% = (45839.3/10000)*100 = 458.39% 

Thus the final velocity of object depends on the initial velocity of the body, mass and radius of planet but independent of mass of object.