Centre of Mass and Collisions

Amount of energy loss (in J) in the collision is:

Two blocks of masses 2 kg and 1 kg respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in 2 s is ________ m. (Take $g=10$ m/s²)

The position of center of mass of three masses 2 kg, 3 kg and 15 kg placed with respect to mid point ($p$) of normal bisector, as shown in the figure is ________.

Two identical bodies A and B of equal masses have initial velocities $\vec v_1=4\hat i$ m/s and $\vec v_2=4\hat j$ m/s respectively. The body A has acceleration $\vec a_1=6\hat i+6\hat j$ m/s² while the acceleration of the other body B is zero. The centre of mass of the two bodies moves in ________ path.

Two radioactive substances A and B of mass numbers 200 and 212 respectively, shows spontaneous α-decay with same $Q$ value of 1 MeV. The ratio of energies of $\alpha$-rays produced by A and B is ________.

Two masses of 3.4 kg and 2.5 kg are accelerated from an initial speed of 5 m/s and 12 m/s, respectively. The distances traversed by the masses in the 5th second are 104 m and 129 m, respectively. The ratio of their momenta after 10 s is $\dfrac x8$. The value of $x$ is ________.

In a scattering experiment, a particle of mass $2m$ collides with another particle of mass $m$, which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation $\theta$ of the heavier particle, as shown in the figure, in radians is:

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency $\nu_1$ and ejects the electron with a kinetic energy of 10 eV. The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency $\nu_2$. The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV. It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV) is _____.

If the collision occurs at time $t_0 = 0$, the value of $v_{\text{cm}}/(a\omega)$ will be _____.

If the collision occurs at time $t_0 = \pi/(2\omega)$, then the value of $4b^2/a^2$ will be _____.

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3h$ from the ground, as shown in the figure. A spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity $\vec u_0 = u_0\hat x$ and falls on the ground at a distance $d$ from the building making an angle $\theta$ with the horizontal. It bounces off with a velocity $\vec v$ and reaches a maximum height $h_1$. The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $1/\sqrt3$. Which of the following statement(s) is(are) correct?

A bar of mass $M = 1.00$ kg and length $L = 0.20$ m is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m = 0.10$ kg is moving on the same horizontal surface with $5.00\ \mathrm{m\,s^{-1}}$ speed on a path perpendicular to the bar. It hits the bar at a distance $L/2$ from the pivoted end and returns back on the same path with speed $v$. After this elastic collision, the bar rotates with an angular velocity $\omega$. Which of the following statement is correct?

Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2\times10^{-8}\ \mathrm{N\,m\,rad^{-1}}$. The angular frequency of the oscillations in $n\times10^{-3}\ \mathrm{rad\,s^{-1}}$. The value of $n$ is _____.

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${}^{16}_{7}\mathrm N + {}^{4}_{2}\mathrm{He} \to {}^{1}_{1}\mathrm H + {}^{19}_{8}\mathrm O$ in a laboratory frame is $n$ (in MeV). Assume that ${}^{16}_{7}\mathrm N$ is at rest in the laboratory frame. The masses of ${}^{16}_{7}\mathrm N$, ${}^{4}_{2}\mathrm{He}$, ${}^{1}_{1}\mathrm H$ and ${}^{19}_{8}\mathrm O$ can be taken to be 16.006 $u$, 4.003 $u$, 1.008 $u$ and 19.003 $u$, respectively, where $1\,u = 930\ \mathrm{MeV}\,c^{-2}$. The value of $n$ is _____.

Two blocks of masses 10 kg and 30 kg are placed on the same straight line with coordinates $(0,0)$ cm and $(x,0)$ cm respectively. The block of 10 kg is moved on the same line through a distance of 6 cm towards the other block. The distance through which the block of 30 kg must be moved to keep the position of centre of mass of the system unchanged is :

What percentage of kinetic energy of a moving particle is transferred to a stationary particle when it strikes the stationary particle of 5 times its mass ?

(Assume the collision to be head-on elastic collision)

A pendulum of length 2 m consists of a wooden bob of mass 50 g. A bullet of mass 75 g is fired towards the stationary bob with a speed $v$. The bullet emerges out of the bob with a speed $\dfrac v3$ and the bob just completes the vertical circle. The value of $v$ is ______ ms$^{-1}$. (if $g=10$ m/s$^2$).

A man of 60 kg is running on the road and suddenly jumps into a stationary trolly car of mass 120 kg. Then, the trolly car starts moving with velocity 2 ms$^{-1}$. The velocity of the running man was ______ ms$^{-1}$, when he jumps into the car.

A body of mass M at rest explodes into three pieces, in the ratio of masses $1:1:2$. Two smaller pieces fly off perpendicular to each other with velocities of 30 ms$^{-1}$ and 40 ms$^{-1}$ respectively. The velocity of the third piece will be :

A bag of sand of mass 9.8 kg is suspended by a rope. A bullet of 200 g travelling with speed 10 ms$^{-1}$ gets embedded in it, then loss of kinetic energy will be :