A particle initially at rest starts moving from reference point. \(\mathrm{x}=0\) along \(\mathrm{x}\)-axis, with velocity \(v\) that varies as \(v=4 \sqrt{\mathrm{x} m} / \mathrm{s}\). The acceleration of the particle is _______ \( \mathrm{ms}^{-2}\).

In an engine the piston undergoes vertical simple harmonic motion with amplitude \(7\, cm\) . A washer rests on top of the piston and moves with it. The motor speed is slowly increased. The frequency of the piston at which the washer no longer stays in contact with the piston, is close to ...... \(Hz\)

The velocity-time graph of an object moving along a straight line is shown in figure. What is the distance covered by the object between \(t=0\) to \(t=4 \mathrm{~s}\) ?

The surface of water in a water tank of cross section area \(750\,cm ^2\) on the top of a house is \(h m\). above the tap level. The speed of water coming out through the tap of cross section area \(500\,mm ^2\) is \(30\,cm / s\). At that instant, \(\frac{d h}{d t}\) is \(x \times 10^{-3} m / s\). The value of \(x\) will be \(.............\).

Escape velocity of a body from earth is \(11.2 \mathrm{~km} / \mathrm{s}\). If the radius of a planet be one-third the radius of earth and mass be one-sixth that of earth, the escape velocity from the plate is _______.

A closed vessel contains \(0.1\) mole of a monoatomic ideal gas at \(200\, K\). If \(0.05\) mole of the same gas at \(400\, K\) is added to it, the final  equilibrium temperature (in \(K\) ) of the gas in the vessel will be closed to

An observer looks at a distant tree of height \(10\ m\) with a telescope of magnifying power of \(20\). To the observer the tree appears :

A bat moving at \(10\,ms^{-1}\) towards a wall sends a sound signal of \(8000\,Hz\) towards it. On reflection it hears a sound of frequency \(f\). The value of \(f\) in \(Hz\) is close to (speed of sound \(= 320\,ms^{-1}\) )

A capacitor is discharging through a resistor \(R\). Consider in time \(t _{1}\), the energy stored in the capacitor reduces to half of its initial value and in time \(t_{2}\), the charge stored reduces to one eighth of its initial value. The ratio \(t_{1} / t_{2}\) will be ................

An alternating voltage of amplitude \(40 \mathrm{~V}\) and frequency \(4 \mathrm{kHz}\) is applied directly across the capacitor of \(12 \mu \mathrm{F}\). The maximum displacement current between the plates of the capacitor is nearly _______.