A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $$m=0.4$$ $$kg$$ is at rest on this surface. An impulse of $$1.0$$ $$Ns$$ is applied to the block at time $$t=0$$ so that it starts moving along the $$x$$-axis with a velocity $$v\left( t \right) = {v_0}{e^{ - t/\tau }},$$ where $${v_0}$$ is a constant and $$\tau = 4s.$$ The displacement of the block, in metres, at $$t = \tau $$ is ______________ Take $${e^{ - 1}} = 0.37.$$

A particle, of mass $${10^{ - 3}}$$ $$kg$$ and charge $$1.0$$ $$C,$$ is initially at rest. At time $$t=0,$$ the particle comes under the influence of an electric field $$\overrightarrow E \left( t \right) = {E_0}\sin \,\,$$ $$\omega t\widehat i,$$ where $${E_0} = 1.0\,N{C^{ - 1}}$$ and $$\omega = 10{}^3\,rad\,{s^{ - 1}}.$$ Consider the effect of only the electrical force on the particle. Then the maximum speed, in $$m{s^{ - 1}},$$ attained by the particle at subsequent times is _______________.

In the List-$${\rm I}$$ below, four different paths of a particle are given as functions of time. In these functions, $$\alpha $$ and $$\beta $$ are positive constants of appropriate dimensions and $$\alpha \ne \beta $$ In each case, the force acting on the particle is either zero or conservative. In List-$${\rm I}{\rm I}$$, five physical quantities of the particle are mentioned $$\overrightarrow p $$ is the linear momentum, $$\overrightarrow L $$ is the angular momentum about the origin, $$K$$ is the kinetic energy, $$U$$ is the potential energy and $$E$$ is the total energy. Match each path in List-$${\rm I}$$ with those quantities in List-$${\rm II}$$, which are conserved for that path.

LIST - I		LIST - II
P.	$$\overrightarrow r $$(t)=$$\alpha $$ $$t\,\widehat i + \beta t\widehat j$$	1.	$$\overrightarrow p $$
Q.	$$\overrightarrow r \left( t \right) = \alpha \cos \,\omega t\,\widehat i + \beta \sin \omega t\,\widehat j$$	2.	$$\overrightarrow L $$
R.	$$\overrightarrow r \left( t \right) = \alpha \left( {\cos \omega t\,\widehat i + \sin \omega t\widehat j} \right)$$	3.	K
S.	$$\overrightarrow r \left( t \right) = \alpha t\,\widehat i + {\beta \over 2}{t^2}\widehat j$$	4.	U
		5.	E

A steel wire of diameter $$0.5$$ $$mm$$ and Young's modulus $$2 \times {10^{11}}\,\,N{m^{ - 2}}$$ carries a load of mass $$M.$$ The length of the wire with the load is $$1.0$$ $$m.A$$ vernier scale with $$10$$ divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count $$1.0$$ $$mm$$ , is attached. The $$10$$ divisions of the vernier scale correspond to $$9$$ divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by $$1.2$$ $$kg,$$ the vernier scale division which coincides with a main scale division is _____________. Take $$g = 10\,m\,{s^{ - 2}}.$$ and $$\pi = 3.2.$$