An object is projected with kinetic energy K from a point A at an angle \(60^{\circ}\) with the horizontal. The ratio of the difference in kinetic energies points B and C to that at point A (see figure), in the absence of air friction is :

In a Young's double slit experiment, three polarizers are kept as shown in the figure. The transmission axes of \(P_1\) and \(P_2\) are orthogonal to each other. The polarizer \(P_3\) covers both the slits with its transmission axis at \(45^{\circ}\) to those of \(P_1\) and \(P_2\). An unpolarized light of wavelength \(\lambda\) and intensity \(I_0\) is incident on \(P_1\) and \(P_2\). The intensity at a point after \(P_3\) where the path difference between the light waves from \(s_1\) and \(s_2\) is \(\frac{\lambda}{3}\), is _______.

An ideal gas at atmospheric pressure is adiabatically compressed so that its density becomes \(32\) times of its initial value. If the final pressure of gas is \(128\) atmosphers, the value of \(\gamma\) the gas is

A ball of mass \(160\, g\) is thrown up at an angle of \(60^o\) to the horizontal at a speed of \(10\, m\,s^{-1}\) . The angular momentum of the ball at the highest point of the trajectcry with respect to the point from which the ball is thrown is nearly ........ \(kg\, m^2/s\)  \((g\, = 10\, m\,s^{-2})\)

Given below are two statements: Statements \(I:\) The temperature of a gas is \(-73^{\circ}\,C\). When the gas is heated to \(527^{\circ}\,C\), the root mean square speed of the molecules is doubled. Statement \(II:\) The product of pressure and volume of an ideal gas will be equal to translational kinetic energy of the molecules. In the light of the above statements, choose the correct answer from the options given below :

A uniform solid cylinder with radius \(R\) and length L has moment of inertia \(I_1\), about the axis of cylinder. A concentric solid cylinder of radius \(R^{\prime}=\frac{R}{2}\) and length \(L^{\prime}=\frac{L}{2}\) is caned out of the original cylinder. If \(I_2\) is the moment of inertia of the carved out portion ot the cylinder then \(\frac{I_1}{I_2}=..........\) (Both \(I_1\) and \(I_2\) are about the axis of the cylinder)

If the time period \(t\) of the oscillation of a drop of liquid of density \(d\), radius \(r\), vibrating under surface tension \(s\) is given by the formula \(t = \sqrt {{r^{2b}}\,{s^c}\,{d^{a/2}}} \) . It is observed that the time period is directly proportional to \(\sqrt {\frac{d}{s}} \) . The value of \(b\) should therefore be

The approximate height from the surface of earth at which the weight of the body becomes \(\frac{1}{3}\) of its weight on the surface of earth is \(..........km\) : [Radius of earth \(R =6400 \,km\) and \(\sqrt{3}=1.732]\)

The amplitude of wave disturbance propagating in the positive \(x\)-direction is given by \(y=\frac{1}{(1+x)^{2}}\) at time \(t=0\) and \(y=\frac{1}{1+(x-2)^{2}}\) at \(t=1\) s, where \(x\) and \(y\) are in metres. The shape of wave does not change during the propagation. The velocity of the wave will be \(...\,{m} / {s}.\)

A proton with a kinetic energy of \(2.0\,eV\) moves into a region of uniform magnetic field of magnitude \(\frac{\pi}{2} \times 10^{-3}\,T\). The angle between the direction of magnetic field and velocity of proton is \(60^{\circ}\). The pitch of the helical path taken by the proton is \(..........cm\) (Take, mass of proton \(=1.6 \times 10^{-27}\,kg\) and Charge on proton \(=1.6 \times 10^{-19}\,kg)\)

A spring of unstretched length \(l\) has a mass \(m\) with one end fixed to a rigid support .Assuming spring to be made of a uniform wire, the kinetic energy possessed by it if its free end is pulled with uniform velocity \(v\) is