If a train travelling at $72\,\, kmph$ is to be brought to rest in a distance of $200$ metres, then its retardation should be............$m{s^{ - 2}}$

A particle starts from origin at $t=0$ with a velocity $5 \hat{i} \mathrm{~m} / \mathrm{s}$ and moves in $x-y$ plane under action of a force which produces a constant acceleration of $(3 \hat{i}+2 \hat{j}) \mathrm{m} / \mathrm{s}^2$. If the $x$-coordinate of the particle at that instant is $84 \mathrm{~m}$, then the speed of the particle at this time is $\sqrt{\alpha} \mathrm{m} / \mathrm{s}$. The value of $\alpha$ is___________.

The velocity-time graphs of a car and a scooter are shown in the figure. $(i)$ the difference between the distance travelled by the car and the scooter in $15\, s$ and $(ii)$ the time at which the car will catch up with the scooter are, respectively

A particle executes the motion described by $x(t) = x_0 (1 - e^{-\gamma t} )$ ; જ્યાં $t\, \geqslant \,0\,,\,{x_0}\, > \,0$.

$(a)$ Where does the particle start and with what velocity ?

$(b)$ Find maximum and minimum values of $x(t),\, v(t)$ $a(t)$. Show that $x(t)$ and $a(t)$ increase with time and $v(t)$ decreases with time.

A particle starts from rest, accelerates at $2 \,ms^{-2}$ for $10\,s$ and then goes for constant speed for $30\,s$ and then decelerates at $ 4\, ms^{-2}$ till it stops. What is the distance travelled by it.........$m$

An object with a mass $10 \,kg$ moves at a constant velocity of $10 \,m/sec$. A constant force then acts for $4\, second$ on the object and gives it a speed of $2\, m/sec$ in opposite direction. The acceleration produced in it, is ........ $m/{\sec ^2}$