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 \(18 \mathrm{~cm}\) 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
\((2 \mathrm{V} \cos \theta-v)\left(v^{2} \sin ^{2} \theta-2 g h\right)^{\mathrm{Uz}}=v \mathrm{V} \sin \theta\)
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 \(1+\dfrac{4 e}{\pi}\).

[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 \(\dfrac{\mu}{\text { (distance) }^{5}}\) 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 \(\dfrac{a}{\sqrt{n^{2}-1}}\).

[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 \(\dfrac{\sqrt{2}}{8}\) 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 \(\mu, \mu^{\prime},\) 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 \(\left(\mu u^{2}+\lambda u^{3}\right)\) and the velocity of projection at distance \(\mathrm{R}\) is \(\mathrm{V}\), show that the particle will ultimately go off to infinity if \(\mathrm{V}^{2}>\dfrac{2 \mu}{\mathrm{R}}+\dfrac{\lambda}{\mathrm{R}^{2}}\)

[13M]