Obtain a relation for the distance travelled by an object moving with a uniform acceleration in the interval between the $4^{th}$ and $5^{th}$ seconds.

$A$ ball is dropped from a height of $20\, m$. $A$ second ball is thrown downwards from the same height after one second with initial velocity $u$. If both the balls reach the ground at the same time, calculate the initial velocity of the second ball. (Take $g = 10\, m s^{-2}$).

$A$ cyclist driving at $5\, m s^{-1}$ picks up a velocity of $10\, m s^{-1}$ over a distance of $50\, m$. Calculate $(i)$ acceleration $(ii)$ time in which the cyclist picks up the above velocity.

$A$ body moving with uniform acceleration has velocities $20 \, m/s$ and $30 \, m/s$ when passing two points $A$ and $B$. Then the velocity midway between $A$ and $B$ is:

$(a)$ Derive the second equation of motion $S = ut + \frac{1}{2}at^2$ graphically,where the symbols have their usual meanings.
$(b)$ $A$ car accelerates uniformly from $18 \text{ km h}^{-1}$ to $36 \text{ km h}^{-1}$ in $5 \text{ s}$. Calculate the acceleration and the distance covered by the car in that time.

Write any two equations of motion for a body having uniform acceleration.