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 moving coil galvanometer has $$50$$ turns and each turn has an area $$2 \times {10^{ - 4}}\,{m^2}.$$ The magnetic field produced by the magnet inside the galvanometer is $$0.02$$ $$T.$$ The torsional constant of the suspension wire is $${10^{ - 4}}\,N\,m\,ra{d^{ - 1}}.$$ When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by $$0.2$$ $$rad$$. The resistance of the coil of the galvanometer is $$50\Omega .$$ This galvanometer is to be converted into an ammeter capable of measuring current in the range $$0-1.0$$ $$A$$. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is _____________.

A ball is projected from the ground at an angle of $${45^o}$$ with the horizontal surface. It reaches a maximum height of $$120$$ $$m$$ and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of $${30^o}$$ with the horizontal surface. The maximium height it reaches after the bounce, in metres, is ______________.

One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is $$100\,K$$ and the universal gas constant $$R=8.0$$ $$J\,mo{l^{ - 1}}{K^{ - 1}},$$ the decrease in its internal energy, in Joule, is ____________.