A particle of mass \(m\) is moving with speed \(2\, v\) collides with a mass \(2\,m\) moving with speed \(v\) in the same direction. After collision, the first mass is stopped completely while the second one splits into two particles each of mass \(m\), which move at angle \(45^o\) with respect to the original direction. The speed of each of the moving particle will be

A parallel-plate capacitor of capacitance \(40 \mu \mathrm{~F}\) is connected to a 100 V power supply. Now the intermediate space between the plates is filled with a dielectric material of dielectric constant \(\mathrm{K}=2\). Due to the introduction of dielectric material, the extra charge and the change in the electrostatic energy in the capacitor, respectively, are

A cube having a side of 10 cm with unknown mass and 200 gm mass were hung at two ends of an uniform rigid rod of 27 cm long. The rod along with masses was placed on a wedge keeping the distance between wedge point and 200 gm weight as 25 cm. Initially the masses were not at balance. A beaker is placed beneath the unknown mass and water is added slowly to it. At given point the masses were in balance and half volume of the unknown mass was inside the water.
(Take the density of unknown mass is more than that of the water, the mass did not absorb water and water density is \(1 \mathrm{gm} / \mathrm{cm}^3\).) The unknown mass is ________ kg.

The period of oscillation of a simple pendulum is \(T=2\pi \sqrt {\frac{l}{g}} \). Measured value of \(L\) is \(20.0\; cm\) known to \(1\; mm\) accuracy and time for \(100\) oscillations of the pendulum is found to be \(90\ s\) using a wrist watch of \(1\; s\) resolution. The accuracy in the determination of \(g\) is   ........ \(\%\)

A particle moves from the point \(\left( {2.0\hat i + 4.0\hat j} \right)\,m\), at \(t = 0\) with an initial velocity \(\left( {5.0\hat i + 4.0\hat j} \right)\,m{s^{ - 1  }}\). It is acted upon by a constant force which produces a constant acceleration \(\left( {4.0\hat i + 4.0\hat j} \right)\,m{s^{ - 2}}\). What is the distance of the particle from the origin at time \(2\,s\)

An electron of mass \(m _{ e }\) and a proton of mass \(m _{ p }=1836 m _{ e }\) are moving with the same speed. The ratio of their de Broglie wavelength \(\frac{\lambda_{\text {electron }}}{\lambda_{\text {proton }}}\) will be ....... .

A solution containing active cobalt \({}_{27}^{60}Co\) having activity of \(0.8\,\mu Ci\) and decay constant \(\lambda \) is injected in an animal's body. If \(1 \,cm^3\) of blood is drawn from the animal's body after \(10\, hrs\) of injection, the activity found was \(300\, decays\) per minute. What is the volume of blood that is flowing in the body?..........\(litres\) ( \(ICi = 3.7 \times 10^{10}\) decay per second and at \(t = 10\, hrs\) \({e^{ - \lambda t}} = 0.84\) )

Let there be three independent events \(E _{1}, E _{2}\) and \(E _{3}\). The probability that only \(E _{1}\) occurs is \(\alpha\), only \(E _{2}\) occurs is \(\beta\) and only \(E _{3}\) occurs is \(\gamma .\) Let \('p'\) denote the probability of none of events occurs that satisfies the equations \((\alpha-2 \beta) p =\alpha \beta\) and \((\beta-3 \gamma) p =2 \beta \gamma .\) All the given probabilities are assumed to lie in the interval \((0,1)\) Then, \(\frac{\text { Probability of occurrence of } E _{1}}{\text { Probability of occurrence of } E _{3}}\) is equal to ..........

A mass \(m\) attached to free end of a spring executes SHM with a period of \(1\; s\). If the mass is increased by \(3\; kg\) the period of oscillation increases by one second, the value of mass \(m\) is \(..............kg\).

With what speed should a galaxy move outward with respect to earth so that the sodium-D line at wavelength \(5890\) \(\stackrel{\circ}{{A}}\) is observed at \(5896\) \(\stackrel{\circ}{{A}}\) ? (in \({km} / {sec}\))

If \((a, b)\) be the orthocentre of the triangle whose vertices are \((1,2),(2,3)\) and \((3,1)\), and \(I_1=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}, \mathrm{I}_2=\int_{\mathrm{a}}^{\mathrm{b}} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}\) , then \(36 \frac{\mathrm{I}_1}{\mathrm{I}_2}\) is equal to :

A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is \(4 \mathrm{~m}\), then the time period of small oscillations will be _______ \(s\). [take \(\mathrm{g}=\pi^2 \mathrm{~ms}^{-2}\)]

If in a \(G.P.\) of \(64\) terms, the sum of all the terms is \(7\) times the sum of the odd terms of the \(G.P,\) then the common ratio of the \(G.P\). is equal to