2004

1) A particle moves with a central acceleration which varies inversely as the cube of the distance, if it be projected from an apse at a distance $a$ from the origin with a velocity $\sqrt{2}$ times the velocity for a circle of radius $a$, determine the path.

[10M]

2) A particle is projected along the inside of a smooth vertical circle of radius $18\mathrm{c}\mathrm{m}$ from the lowest point. Find the velocity of projection so that after leaving the circle, the particle may pass through the centre.

[14M]

2003

1) A particle moves with an acceleration which is always towards, and equal to $\mu$ divided by the distance from a fixed point $O$. If it starts from rest at a distance $a$ from $O$ ,show that it will arrive at $O$, in time

[10M]

2) A shell fired with velocity $V$ at an elevation $\theta$ hits an airship at a height $h$ from the ground, which is moving horizontally away from the gun with velocity $v$. Show that if
$(2\mathrm{V}\cos \theta -v){(v^2{\sin }^2\theta -2gh)}^{\mathrm{U}\mathrm{z}}=v\mathrm{V}\sin \theta$
the shell might have also hit the ship if the latter had remained stationary in the position it occupied when the gun was actually fired.

[10M]

TBC

3) Assuming the eccentricity e of a planet’s erbitis a small fraction, show that the ratio of the time taken by the planet to travel over the balves of its orbit separated by the minor axis is nearly $1+{\displaystyle \frac{4e}{\pi }}$.

[10M]

2002

1) A particle falls towards the earth from infinity. Show that its velocity on reaching the earth is the same as it would have acquired in falling with constant accelcration $g$ through a distance equal to the earth’s radius.

[10M]

2) A particle moves with a central acceleration ${\displaystyle \frac{\mu }{{\text{ (distance) }}^5}}$ and is projected from an apse at a distance $a$ with velocity equal to $a$ times that would be acquired in falling from infinity. Show that the other apsidal distance is ${\displaystyle \frac{a}{\sqrt{n^2-1}}}$.

[10M]

63 If a planet were suddenly stopped in its orbit, supposed circular, show that it would fall into the sun in a time which is ${\displaystyle \frac{\sqrt{2}}{8}}$ times the period of the planet’s revolution.

[10M]

2001

1) A particle rests in equilibrium under the reaction of two centers of forces which attract directly as the distance, their intensity being $\mu ,{\mu }^{\mathrm{\prime }},$ the particle is displaced slightly towards one of them, show that the time of a small oscillation is

[10M]

2) A particle moves with central acceleration $(\mu u^2+\lambda u^3)$ and the velocity of projection at distance $\mathrm{R}$ is $\mathrm{V}$, show that the particle will ultimately go off to infinity if ${\mathrm{V}}^2>{\displaystyle \frac{2\mu }{\mathrm{R}}}+{\displaystyle \frac{\lambda }{{\mathrm{R}}^2}}$

[13M]