Moving body approaching Neutron Star

MOVING BODY APPROACHING NEUTRON STAR 

INTRODUCTION

The Neutron Star is a highly compact star primarily composed of neutrons. It has density the same as that of an atom. A Neutron star is born from a supernova explosion. Consider a body of mass m moving with an initial speed of 10 km/s approaching the star at a distance of 4R from its surface. As the body enters Neutron Star’s gravitational field, it begins accelerating. This type of acceleration is continuous acceleration as it's continuously being accelerated due to Neutron Star’s gravity.

CALCULATION

The final velocity of the moving body when it touches Neutron Star’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 Neutron Star [R = 11000m]

M - Mass of Neutron Star, M = 2*2*10³⁰ kg

h - Height above the surface of Neutron Star, 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 + [3.88*10¹⁶] 

v² = 3.88*10¹⁶

v    = 196,977,156 m/s

CONCLUSION

Thus the final velocity of the body on the surface of Neutron Star is 196,977,156 m/s or if it's moving with an initial velocity of 10km/s. It depends on the initial velocity of the body, mass and radius of planet but independent of mass of object.

Time dilation in TGV train

TIME DILATION IN TGV

INTRODUCTION

TGV [Train a grande vitesse] is a high speed train in France. The TGV has a top speed of 323.75 mph, Consider an observer in TGV who travels from Paris to suburbs and another observer who is stationary with respect to the TGV. According to the Special Theory of Relativity, the TGV’s clock would run slower compared to the observer’s clock. We’ll find the time gained by the TGV relative to the stationary observer.

ASSUMPTION

The effect of Gravitational time dilation is negligible.

CALCULATION

The TGV’s velocity is,

v = 323.75 mph = 143.88 m/s

According to the Special Theory of Relativity, the time dilation equation is,

t' = t/γ [s]

t’ – Actual time or TGV’s time. [s]

t - Proper time or Stationary observer’s time. [s]

γ – Relativistic gamma factor, γ = 1/√ [1-(v/c) ²]

c - Velocity of light [c = 3*10⁸ m/s]

t' = t*√ [1-(v/c) ²]

t' = t*√ [1-2.3*10^-13]

t' = t*√ [0.99999999999977]

t' = t* 0.999999999999885

CONCLUSION

We can observe that proper and actual time isn’t the same which proves that time dilates on TGV relative to the stationary observer. We’ll consider 3 different t’ values and calculate t value. The larger the t’ the more is the difference between t and t’. Thus the TGV gains 0.41nanosecond in 1 hour and 2.1nanosecond in 5 hours over the stationary observer.

Time	t’ [Stationary observer] (s)	t [Observer in TGV] (s)	Difference (s)
1 minute	60	59.9999999999931	0.0000000000069
1 hour	3600	3599.99999999959	0.00000000041
5 hours	18000	17999.9999999979	0.0000000021