A particle of mass $0.1 \,kg$ is subjected to a force which varies with distance as shown in fig. If it starts its journey from rest at $x = 0$, its velocity at $x = 12\,m$ is .......... $m/s$

A particle of mass $m$ with initial kinetics energy $K$ approaches the origin from $x =+\infty$. Assume that a conservative force acts on it and its potential energy $V ( x )$ is given by $V ( x )=\frac{ K }{\exp \left(3 x / x _0\right)+\exp \left(-3 x / x _0\right)}$ where, $x_0=1 m$. The speed of the particle at $x =0$ is

A space craft of mass $M$ is moving with velocity $V$ and suddenly explodes into two pieces. A part of it of mass m becomes at rest, then the velocity of other part will be

A particle of mass m moving with velocity ${V_0}$ strikes a simple pendulum of mass $m$ and sticks to it. The maximum height attained by the pendulum will be

A balloon filled with helium rises against gravity increasing its potential energy. The speed of the balloon also increases as it rises. How do you reconcile this with the law of conservation of mechanical energy ? You can neglect viscous drag of air and assume that density of air is constant.

A particle $(\mathrm{m}=1\; \mathrm{kg})$ slides down a frictionless track $(AOC)$ starting from rest at a point $A$ (height $2\; \mathrm{m}$ ). After reaching $\mathrm{C}$, the particle continues to move freely in air as a projectile. When it reaching its highest point $P$ (height $1 \;\mathrm{m}$ ). the kinetic energy of the particle (in $\mathrm{J}$ ) is : (Figure drawn is schematic and not to scale; take $\left.g=10 \;\mathrm{ms}^{-2}\right)$