If the velocity of a body related to displacement \({x}\) is given by \(v=\sqrt{5000+24 {x}} \;{m} / {s}\), then the acceleration of the body is \(\ldots \ldots {m} / {s}^{2}\)

A particle of mass \(\mathrm{m}\) is dropped from a height \(h\) above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of \(\sqrt{2 \mathrm{gh}} .\) If they collide head-on completely inelastically, the time taken for the combined mass to reach the ground, in units of \(\sqrt{\frac{\mathrm{h}}{\mathrm{g}}}\) is

A diver looking up through the water sees the outside world contained in a circular horizon. The refractive index of water is \(\frac{4}{3}\),  and the diver's eyes are \(15\, cm\) below the surface of water. Then the radius of the circle is

Statement \(I :\) The ferromagnetic property depends on temperature. At high temperature, ferromagnet becomes paramagnet. Statement \(II :\) At high temperature, the domain wall area of a ferromagnetic substance increases. In the light of the above statements, choose the most appropriate answer from the options given below :

The Young's modulus of steel wire of radius \(r\) and length \(L\) is \(Y\). If the radius \(r\) and length \(L\) of the wire are doubled then the value of \(Y\)

A plane electromagnetic wave propagating in \(\mathrm{x}\)-direction is described by \(\mathrm{E}_{\mathrm{y}}=\left(200\  \mathrm{Vm}^{-1}\right) \sin \left[1.5 \times 10^7 \mathrm{t}-0.05\  \mathrm{x}\right] \text {; }\) The intensity of the wave is :(Use \(\epsilon_0=8.85 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}\) )

A wind-10.Powered generator converts wind energy into electrical energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. For wind speed \(v\), the electrical 10.Power output will be most likely proportional to

A long solenoid of radius \(\mathrm{R}\) carries a time \((t)-\)dependent current \(\mathrm{I}(\mathrm{t})=\mathrm{I}_{0} \mathrm{t}(1-\mathrm{t})\). A ring of radius \(2 \mathrm{R}\) is placed coaxially near its middle. During the time interval \(0 \leq t \leq 1\), the induced current \(\left(\mathrm{I}_{\mathrm{R}}\right)\) and the induced \(\mathrm{EMF}\left(\mathrm{V}_{\mathrm{R}}\right)\) in the ring change as

The electrostatic potential in a charged spherical region of radius \(r\) varies as \(V=a r^3+b\), where \(a\) and b are constants. The total charge in the sphere of unit radius is \(\alpha \times \pi a \in_0\). The value of \(\alpha\) is _________ .

A bullet of mass \(20\,g\) has an initial speed of \(1\,ms^{-1}\) just before it starts penetrating a mud wall of thickness \(20\,cm.\) If the wall offers a mean resistances of \(2.5 \times 10^{-2}\,N,\) the speed of the bullet after emerging from the other side of the wall is close to .............. \(\mathrm{ms}^{-1}\)

A particle of mass \(m\) is moving along a trajectory given by
\(x = x_0 + a\, cos\,\omega_1 t\)
\(y = y_0 + b\, sin\,\omega_2t\)
The torque, acing on the particle about the origin, at \(t = 0\) is