We will cover following topics

Rectilinear Motion

Rectilinear motion is another name for straight-line motion. The following four equations hold true for such motion:

For position of the moving object:

\[x_{2}-x_{1}=v_{1} t+1 / 2 a t^{2}\]

For final velocity:

\[v_{2}=v_{1}+a t\]

For displacement:

\[\Delta d=v_{1} t+1 / 2 a t^{2}\]

For final velocity, when time is not given:

\[v_{2}^{2}=v_{1}^{2}+2 a(\Delta d)\]

PYQs

Rectilinear Motion

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 $\mu\left(\dfrac{a^{5}}{x^{2}}\right)^{1 / 3}$ 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 $'a'$ 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

\[\mu \left( x+ \dfrac{a^4}{x^3} \right)\]

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 $\dfrac{a}{2}$.

[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 $mn^2x$ towards a point $O$ on the line, where $x$ is the distance from $O$. If $x=a$ and $\dfrac{dx}{dt}=u$ 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 $m\left(x+\dfrac{a^{4}}{x^{3}}\right)$ towards the origin. If it starts from rest at a distance $a$, show that it will arrive at origin in time $\dfrac{\pi}{4}$.

[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 $m\left(x+\dfrac{a^{4}}{x^{3}}\right)$ 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]

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