Centre of Mass and Collisions

Three identical spheres each of mass M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 3 m each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be $\sqrt x$ m. The value of $x$ is ______.

As per the given figure, two blocks each of mass 250 g are connected to a spring of spring constant 2 Nm$^{-1}$. If both are given velocity $v$ in opposite directions, then maximum elongation of the spring is :

A body of mass 8 kg and another of mass 2 kg are moving with equal kinetic energy. The ratio of their respective momenta will be :

A nucleus of mass M at rest splits into two parts having masses $\dfrac{M'}3$ and $\dfrac{2M'}3$ $(M'<M)$. The ratio of de Broglie wavelength of two parts will be :

Sand is being dropped from a stationary dropper at a rate of 0.5 kgs$^{-1}$ on a conveyor belt moving with a velocity of 5 ms$^{-1}$. The power needed to keep the belt moving with the same velocity will be :

A mass 0.9 kg, attached to a horizontal spring, executes SHM with an amplitude $A_1$. When this mass passes through its mean position, then a smaller mass of 124 g is placed over it and both masses move together with amplitude $A_2$. If the ratio $\dfrac{A_1}{A_2}$ is $\dfrac\alpha{\alpha-1}$, then the value of $\alpha$ will be ______.

A balloon has mass of 10 g in air. The air escapes from the balloon at a uniform rate with velocity 4.5 cm/s. If the balloon shrinks in 5 s completely. Then, the average force acting on that balloon will be (in dyne).

The distance of centre of mass from end A of a one dimensional rod (AB) having mass density $\rho=\rho_0\left(1-\dfrac{x^2}{L^2}\right)$ kg/m and length L (in meter) is $\dfrac{3L}\alpha$ m. The value of $\alpha$ is ______. (where $x$ is the distance from end A)

Two bodies of mass 1 kg and 3 kg have position vectors $\hat i+2\hat j+\hat k$ and $-3\hat i-2\hat j+\hat k$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :

A bullet of mass 10 gram is fired with velocity of 100 m/s from the gun of mass 1 kg. Recoil speed of the gun is :

A body moving horizontally on a smooth surface with a speed of 20 m s$^{-1}$ splits into two parts and continue to move in the same direction. The masses of two parts are in the ratio of 1:2. The smaller part moves with a speed of 30 m s$^{-1}$. The fractional change is the kinetic energy is given by $\dfrac1x$. The value of $x$ is _______

The value of $t$ is ____.

The value of $x$ is ____.

A heavy nucleus $N$, at rest, undergoes fission $N \to P + Q$, where $P$ and $Q$ are two lighter nuclei. Let $\delta = M_N - M_P - M_Q$, where $M_P$, $M_Q$ and $M_N$ are the masses of $P$, $Q$ and $N$, respectively. $E_P$ and $E_Q$ are the kinetic energies of $P$ and $Q$, respectively. The speeds of $P$ and $Q$ are $v_P$ and $v_Q$, respectively. If $c$ is the speed of light, which of the following statement(s) is(are) correct?

Two stars of masses m and 2m at a distance d rotate about their common centre of mass in free space. The period of revolution is :

In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillation will be :

A ball with a speed of 9 m/s collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of 30° with the original direction. The ratio of velocities of the balls after collision is $x:y$, where $x$ is _______ .

A rod of mass $m$ and length $L$, pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed $v$ strikes the rod horizontally at a distance $x$ from its pivoted end and gets embedded in it. The combined system now rotates with angular speed $\omega$ about the pivot. The maximum angular speed $\omega_M$ is achieved for $x = x_M$. Then

A small particle of mass $m$ moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $L = L_0$ the particle speed is $v = v_0$. The piston is moved inward at a very low speed $V$ such that $V \ll \dfrac{dL}{L}v_0$, where $dL$ is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

Suppose a $\ce{^{226}_{88}Ra}$ nucleus at rest and in ground state undergoes α-decay to a $\ce{^{222}_{86}Rn}$ nucleus in its excited state. The kinetic energy of the emitted α particle is found to be 4.44 MeV. $\ce{^{222}_{86}Rn}$ nucleus then goes to its ground state by γ-decay. The energy of the emitted γ photon is ______ keV.

[Given: atomic mass of $\ce{^{226}_{88}Ra}$ = 226.005 u, atomic mass of $\ce{^{222}_{86}Rn}$ = 222.000 u, atomic mass of α particle = 4.000 u, 1 u = 931 MeV/c², c is speed of the light]