Projectiles \(A\) and \(B\) are thrown at angles of \(45^{\circ}\) and \(60^{\circ}\) with vertical respectively from top of a \(400 \mathrm{~m}\) high tower. If their ranges and times of flight are same, the ratio of their speeds of projection \(v_A: v_B\) is _______.

Two coils have mutual inductance \(0.002 \ \mathrm{H}\). The current changes in the first coil according to the relation \(\mathrm{i}=\mathrm{i}_0 \sin \omega \mathrm{t}\), where \(\mathrm{i}_0=5 \mathrm{~A}\) and \(\omega=50 \pi\) \(\mathrm{rad} / \mathrm{s}\). The maximum value of \(\mathrm{emf}\) in the second coil is \(\frac{\pi}{\alpha} \mathrm{V}\). The value of \(\alpha\) is_______.

A body is projected at \(t = 0\) with a velocity \(10\,ms^{-1}\) at an angle of \(60^o\) with the horizontal. The radius of curvature of its trajectory at \(t=1\,s\) is \(R.\) Neglecting air resistance and taking acceleration due to gravity \(g = 10\,ms^{-2},\) the radius of \(R\) is   ........ \(m\)

A galvanometer has a \(50\) \(division\) scale. Battery has no internal resistance. It is found that there is deflection of \(40\) \(divisions\) when \(R = 2400\,\Omega \). Deflection becomes \(20\) \(divisions\) when resistance taken from resistance box is \(4900\,\Omega \) . Then we can conclude

The current flowing through an ac circuit is given by \(I=5 \sin (120 \pi t) A\) How long will the current take to reach the peak value starting from zero?

The equations of two waves are given by \(y_{1}=5 \sin 2 \pi(x-v t) \,c m\,\) \(y_{2}=3 \sin 2 \pi(x-v t+1.5) \,c m\) These waves are simultaneously passing through a string. The amplitude of the resulting wave is.........\(cm\)

A point charge +q is placed at the origin. A second point charge +9 q is placed at \((\mathrm{d}, 0,0)\) in Cartesian coordinate system. The point in between them where the electric field vanishes is :

Two separate wires \(A\) and \(B\) are stretched by \(2 \,mm\) and \(4\, mm\) respectively, when they are subjected to a force of \(2\, N\). Assume that both the wires are made up of same material and the radius of wire \(B\) is 4 times that of the radius of wire \(A\). The length of the wires \(A\) and \(B\) are in the ratio of \(a : b\). Then \(a / b\) can be expressed as \(1 / x\) where \(x\) is

In single slit diffraction pattern, the wavelength of light used is \(628\) nm and slit width is \(0.2\) mm, the angular width of central maximum is \(\alpha \times 10^{-2}\) degrees. The value of \(\alpha\) is _______.

Using Bohr's model, calculate the ratio of the magnetic fields generated due to the motion of the electrons in the \(2^{\text{nd}}\) and \(4^{\text{th}}\) orbits of hydrogen atom _______.

The relative error in the determination of the surface area of a sphere is \(\alpha \). Then the relative error in the determination of its volume is