PYQs

1) A particle moves in a straight line. Its acceleration is directed towards a fixed point $O$ in the line and is always equal to $\mu {\left({\displaystyle \frac{a^5}{x^2}}\right)}^{1/3}$ when it is at a distance $x$ from $O$. If it starts from rest at a distance $a$ from $O$, then find the time the particle will arrive at $O$.

[2016, 15M]

2) A mass starts from rest at a distance ${}^{\prime }{a}^{\prime }$ from the centre of force which attracts inversely as the distance. Find the time of arriving at the centre.

[2015, 13M]

3) A particle moves with an acceleration

towards the origin. It it starts from rest at a distance $a$ from the origin, find its velocity when its distance from the origin is ${\displaystyle \frac{a}{2}}$.

[2012, 12M]

4) The velocity of a train increases from 0 to $v$ at a constant acceleration $f_1$, then remains constant for an interval and again decreases to 0 at a constant retardation $f_2$. If the total distance described is $x$, find the total time taken.

[2011, 10M]

6) A particle of mass $m$ moves on straight line under an attractive force $m{n}^{2}x$ towards a point $O$ on the line, where $x$ is the distance from $O$. If $x=a$ and ${\displaystyle \frac{dx}{dt}}=u$ when $t=0$, find $x(t)$ for any time $t>0$.

[2011, 10M]

7) After a ball has been falling under gravity for 5 seconds, it passes through a plane of glass and loses half its velocity. If it now reaches the ground in 1 second, find the height of glass above the ground.

[2011, 10M]

8) A particle falls from rest under gravity in a medium whose resistance varies as the velocity of the particle. Find the distance fallen by the particle and its velocity at time $t$.

[2007, 12M]

9) A particle, whose mass is $m$, is acted upon by a force $m(x+{\displaystyle \frac{a^4}{x^3}})$ towards the origin. If it starts from rest at a distance $a$, show that it will arrive at origin in time ${\displaystyle \frac{\pi }{4}}$.

[2006, 15M]

10) A point moving with uniform acceleration describes distances $s_1\$and\$s_2$ metres in successive intervals of time $t_1$ and $t_2$ seconds. Express the acceleration in terms of $s_1$, $s_2$, $t_1$ and $t_2$.

[2004, 12M]

11) A particle of mass $m$ is acted upon by a force $m(x+{\displaystyle \frac{a^4}{x^3}})$ towards the origin. If it starts from rest at a distance a from the origin, show that the time taken by it to reach the origin is $\pi /4$.

[2002, 12M]