# A projectile fired vertically upwards with a speed v escapes from the earth. If it is to be fired at 45$°$ to the horizontal, what should be its speed so that it escapes from the earth? 1.  v 2.  $\frac{\mathrm{v}}{\sqrt{2}}$ 3.  $\sqrt{2}\mathrm{v}$ 4.  2v

Subtopic:  Escape velocity |
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Kepler's second law regarding constancy of the areal velocity of a planet is a consequence of the law of conservation of:

1. Energy

2. Linear momentum

3. Angular momentum

4. Mass

Subtopic:  Kepler's Laws |
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A point $$P$$ lies on the axis of a ring of mass $$M$$ and radius $$a$$ at a distance $$a$$ from its centre $$C$$. A small particle starts from $$P$$ and reaches $$C$$ under gravitational attraction. Its speed at $$C$$ will be:
1. $$\sqrt{\frac{2 GM}{a}}$$
2. $$\sqrt{\frac{2 GM}{a} \left(1 - \frac{1}{\sqrt{2}}\right)}$$
3. $$\sqrt{\frac{2 GM}{a} \left(\sqrt{2} - 1\right)}$$
4. zero

Subtopic:  Gravitational Potential Energy |
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Magnitude of potential energy ($$U$$) and time period ($$T$$) of a satellite are related to each other as:
1. $$T^2\propto \frac{1}{U^{3}}$$
2. $$T\propto \frac{1}{U^{3}}$$
3. $$T^2\propto U^3$$
4. $$T^2\propto \frac{1}{U^{2}}$$

Subtopic:  Satellite |
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Two bodies of masses m and 4m are placed at a distance r. The gravitational potential at a point on the line joining them where the gravitational field is zero is

1.  $-\frac{5\mathrm{Gm}}{\mathrm{r}}$

2.  $-\frac{6\mathrm{Gm}}{\mathrm{r}}$

3.  $-\frac{9\mathrm{Gm}}{\mathrm{r}}$

4.  0

Subtopic:  Gravitational Field |
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If $$A$$ is the areal velocity of a planet of mass $$M,$$ then its angular momentum is:

 1 $$\frac{M}{A}$$ 2 $$2MA$$ 3 $$A^2M$$ 4 $$AM^2$$
Subtopic:  Kepler's Laws |
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In planetary motion, the areal velocity of the position vector of a planet depends on the angular velocity $$(\omega)$$ and the distance of the planet from the sun $$(r)$$. The correct relation for areal velocity is:
1. $$\frac{dA}{dt}\propto \omega r$$
2. $$\frac{dA}{dt}\propto \omega^2 r$$
3. $$\frac{dA}{dt}\propto \omega r^2$$
4. $$\frac{dA}{dt}\propto \sqrt{\omega r}$$

Subtopic:  Kepler's Laws |
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A planet is moving in an elliptical orbit. If $$T, V, E,$$ and $$L$$ stand, respectively, for its kinetic energy, gravitational potential energy, total energy and angular momentum about the center of the orbit, then:
 1 $$T$$ is conserved 2 $$V$$ is always positive 3 $$E$$ is always negative 4 the magnitude of $$L$$ is conserved but its direction changes continuously
Subtopic:  Satellite |
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The gravitational potential difference between the surface of a planet and 10 m above is 5 J/kg. If the gravitational field is supposed to be uniform, the work done in moving a 2 kg mass from the surface of the planet to a height of 8 m is

1.  2J

2.  4J

3.  6J

4.  8J

Subtopic:  Gravitational Potential |
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A projectile is fired upwards from the surface of the earth with a velocity $$kv_e$$ where $$v_e$$ is the escape velocity and $$k<1$$. If $$r$$ is the maximum distance from the center of the earth to which it rises and $$R$$ is the radius of the earth, then $$r$$ equals:
1. $$\frac{R}{k^2}$$
2. $$\frac{R}{1-k^2}$$
3. $$\frac{2R}{1-k^2}$$
4. $$\frac{2R}{1+k^2}$$

Subtopic:  Escape velocity |
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