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Class 10 Class 12
The escape speed of a projectile on the earth’s surface is 11.2 km s–1. A body is projected out with thrice this speed. What is the speed of the body far away from the earth? Ignore the presence of the sun and other planets.

Escape velocity of a projectile from the Earth, vesc = 11.2 km/s

Projection velocity of the projectile, vp = 3vesc

Mass of the projectile =

Velocity of the projectile far away from the Earth = v

Total energy of the projectile on the Earth = mvp2 - mvesc

Gravitational potential energy of the projectile far away from the Earth is zero.

Total energy of the projectile far away from the Earth = mvf

From the law of conservation of energy, we have

mvp2 - mvesc2  =   mvf
vf = ( vp2 - vesc)1/2

= [ (3vesc)2 - vesc2 ]1/2

= √8 vesc

= √8 × 11.2

=  31.68 km/s.
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What is the celestial sphere?

At night, if we see the planets and the stars in the sky, all appear to lie in the hemisphere (rest of the hemisphere we are unable to see because of being on the other side of the earth). This sphere is called the celestial sphere.
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What is Geocentric theory?

According to the geocentric theory, all the astronomical bodies like the moon, the sun and stars revolve around the earth, and the earth is at the centre of the universe.
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What is Heliocentric theory?

According to the Heliocentric theory, the sun is at the centre and various planets revolve around the sun at their axis.
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The position co-ordinates of two particles of masses m1 and m2are (x1, y1, z1) and (x2, y2, z2) respectively. Find the coordinates of the centre of mass.

The position vectors of masses m1 and m2 are respectively,

Let the position coordinates of the centre of mass be (X, Y, Z).

Therefore the position vector of centre of mass is,

Since,

Comparing the coefficients of  , we get

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Is Geodesic always a straight line?

No, Geodesic is a straight line if and only if,  the two points lie on the flat surface. If the two points lie on the curved surface then it is a curved line.
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