Moving Body approaching Alpha Arae

MOVING BODY APPROACHING ALPHA ARAE 

INTRODUCTION

The Alpha Arae is a distant star in our Solar System with a distance of 270 light years from the Earth. Consider a body of mass m moving with an initial speed of 10 km/s approaching Alpha arae at a distance of 4R from its surface. As the body enters Alpha arae’s gravitational field, it begins accelerating. This type of acceleration is continuous acceleration as it's continuously being accelerated due to Alpha arae’s gravity.

CALCULATION

The final velocity of the moving body when it touches Alpha arae’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 Alpha arae [R = 3,130,650,000m]

M - Mass of Alpha arae, M = 1.92 x 10³¹ kg

h - Height above the surface of Alpha arae, h = 4R

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

v - Final velocity of the mass

v² = u² + 2GM [(1/R) – (1/R+4R)]

v² = u² + 2GM [(1/R) – (1/5R)]

v² = u² + 2GM [4R/5R²]

v² = u² + GM [8/5R]

Substituting all values we get,

v² = 10⁸ + [6.545*10¹¹]

v² = 6.546*10¹¹

v    = 809,073.54 m/s

CONCLUSION

Thus the final velocity of the body on the surface of Alpha arae is 809,073.54 m/s if it's moving with an initial velocity of 10km/s.

The change [increase] in velocity is,

 Δ = v – u

 Δ = 809,073.54 – 10000

 Δ = 799,073.54 m/s

The percentage increase in velocity is,

∆% = (799,073/10000)*100 = 7990.73%

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.