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1.

Let the speed of the planet at the perihelion \(P\) in figure shown below be \(v_{_P}\) and the Sun-planet distance \(\mathrm{SP}\) be \(r_{_P}.\) Relation between \((r_{_P},~v_{_P})\) to the corresponding quantities at the aphelion \((r_{_A},~v_{_A})\) is:

1.	\(v_{_P} r_{_P} =v_{_A} r_{_A}\)	2.	\(v_{_A} r_{_P} =v_{_P} r_{_A}\)
3.	\(v_{_A} v_{_P} = r_{_A}r_{_P}\)	4.	none of these

2. Let the speed of the planet at the perihelion \(P\) in the figure shown below be \(v_p\) and the Sun-planet distance \(SP\) be \(r_p.\)Will the planet take equal time to traverse \(BAC\) and \(CPB?\)

          

1.	no	2.	yes
3.	depends on the mass of the planet	4.	we can't say anything

3.

Assuming the earth to be a sphere of uniform mass density, how much would a body weigh halfway down to the centre of the earth if it weighed \(250\) N on the surface?

1.	\(250\) N	2.	\(125\) N
3.	\(175\) N	4.	\(145\) N

4. A \(400~\text{kg}\) satellite is in a circular orbit of radius \(2R_E\) about the Earth. What are the changes in the kinetic and potential energies respectively to transfer it to a circular orbit of radius \(4R_{E}.\) (where \(R_E\) is the radius of the Earth)
1. \(3.13\times 10^{9}~\text{J}~\text{and}~6.25\times10^{9}~\text{J}\)
2. \(3.13\times 10^{9}~\text{J}~\text{and}~-6.25\times10^{9}~\text{J}\)
3. \(-3.13\times 10^{9}~\text{J}~\text{and}~-6.25\times10^{9}~\text{J}\)
4. \(-3.13\times 10^{8}~\text{J}~\text{and}~-6.25\times10^{8}~\text{J}\)

5.

Two uniform solid spheres of equal radii \({R},\) but mass \({M}\) and \(4M\) have a centre to centre separation \(6R,\) as shown in the figure. The two spheres are held fixed. A projectile of mass \(m\) is projected from the surface of the sphere of mass \(M\) directly towards the centre of the second sphere. The expression for the minimum speed \(v\) of the projectile so that it reaches the surface of the second sphere is:

1.	\(\left(\dfrac{3 {GM}}{5 {R}}\right)^{1 / 2}\)	2.	\(\left(\dfrac{2 {GM}}{5 {R}}\right)^{1 / 2}\)
3.	\(\left(\dfrac{3 {GM}}{2 {R}}\right)^{1 / 2}\)	4.	\(\left(\dfrac{5 {GM}}{3 {R}}\right)^{1 / 2}\)

6.

A Saturn year is \(29.5\) times the Earth year. How far is Saturn from the sun if the Earth is \(1.5 \times 10^8 ~\text{km}\) away from the sun?
1. \( 2.01 × 10^{12}~\text{m}\)
2. \(3.86​​ × 10^{12}~\text{m}\)
3. \(1.43 × 10^{12}~\text{m}\)
4. \(4.19​​ × 10^{12}~\text{m}\)

 

7.

A body weights \(63\) N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth?
1. \(98~\text N\) 
2. \(35~\text N\) 
3. \(63~\text N\) 
4. \(28~\text N\) 

8.

A rocket is fired vertically with a speed of \(5\) km/s from the earth’s surface. How far from the earth does the rocket go before returning to the earth?
1. \(8\times10^6\) m
2. \(1.6\times10^6\) m
3. \(6.4\times10^6\) m
4. \(12\times10^6\) m

9.

The escape speed of a projectile on the Earth’s surface is \(11.2 ~\text{Km/s}.\) 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.)
1. \(32.7 ~\text{Km/s}\)
2. \(11.2 ~\text{Km/s}\)
3. \(31.7 ~\text{Km/s}\)
4. \(21.2 ~\text{Km/s}\)