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Class 10 Class 12
A rocket is fired ‘vertically’ from the surface of mars with a speed of 2 km s–1. If 20% of its initial energy is lost due to Martian atmospheric resistance, how far will the rocket go from the surface of mars before returning to it? Mass of mars = 6.4× 1023 kg; radius of mars = 3395 km; G = 6.67× 10-11 N m2 kg–2.

Initial velocity of the rocket, v = 2 km/s = 2 × 103 m/s

Mass of Mars, M = 6.4 × 1023 kg

Radius of Mars, R = 3395 km = 3.395 × 106 m

Universal gravitational constant, G = 6.67× 10–11 N m2 kg–2

Mass of the rocket =

Initial kinetic energy of the rocket = mv

Initial potential energy of the rocket =

Total initial energy = mv2 -

If 20 % of initial kinetic energy is lost due to Martian atmospheric resistance, then only 80 % of its kinetic energy helps in reaching a height.

Total initial energy available =
= (80/100) × (1/2)
mv2 -

=  0.4
mv2 -

Maximum height reached by the rocket =

At this height, the velocity and hence, the kinetic energy of the rocket will become zero.

Total energy of the rocket at height h =

Applying the law of conservation of energy for the rocket,

0.4mv2 -   =

0.4v2 =

=

=

h =

=

=

=

=

= 495 × 103 m

= 495 km.
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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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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 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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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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