The initial velocity of a body moving along a straight line is $7 \, m/s$. It has a uniform acceleration of $4 \, m/s^2$. The distance covered by the body in the $5^{th}$ second of its motion is .......... $m$.

$A$ body of mass $10 \, kg$ is moving with a constant velocity of $10 \, m/s$. When a constant force acts for $4 \, s$ on it,it moves with a velocity of $2 \, m/s$ in the opposite direction. The acceleration produced in it is .......... $m/s^2$.

$A$ body is moving with a uniform velocity of $8\,m/s$. When the body just crosses another body,the second body starts from rest and moves with a uniform acceleration of $4\,m/s^2$. The time after which the two bodies meet will be $...........\,s$.

$A$ particle starts from rest at $x=0 \, m$ with an acceleration of $1 \, m/s^2$. At $t = 5 \, s$,it receives an additional acceleration in the same direction as its motion. At $t = 10 \, s$,its speed and position are $v$ and $x$,respectively. Had the additional acceleration not been provided,its speed and position would have been $v_0$ and $x_0$,respectively. It is found that $x - x_0 = 12.5 \, m$. Then one can conclude that $v - v_0$ is .............. $m/s$.

$A$ particle is moving in a straight line. The variation of position $x$ as a function of time $t$ is given as $x = (t^3 - 6t^2 + 20t + 15) \ m$. The velocity of the body when its acceleration becomes zero is ........... $m/s$.

Derive the equations of motion for constant acceleration using the method of calculus.