Centre of Mass and Collisions

A photon of wavelength $\lambda$ is scattered from an electron, which was at rest. The wavelength shift $\Delta\lambda$ is three times of $\lambda$ and the angle of scattering $\theta$ is 60°. The angle at which the electron recoiled is $\phi$. The value of $\tan\phi$ is : (electron speed is much smaller than the speed of light)

Three masses m, 2m and 3m are moving in $x$-$y$ plane with speed 3u, 2u, and u respectively as shown in figure. The three masses collide at the same point at P and stick together. The velocity of resulting mass will be :

A 4 g bullet is fired horizontally with a speed of 300 m/s into 0.8 kg block of wood at rest on a table. If the coefficient of friction between the block and the table is 0.3, how far will the block slide approximately ?

Two hypothetical planets of masses $m_1$ and $m_2$ are at rest when they are infinite distance apart. Because of the gravitational force they move towards each other along the line joining their centres. What is their speed when their separation is 'd' ?

(Speed of $m_1$ is $v_1$ and that of $m_2$ is $v_2$)

A particle of mass $m$ is projected from the ground with an initial speed $u_0$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_0$. The angle that the composite system makes with the horizontal immediately after the collision is

A bob of mass $m$, suspended by a string of length $l_1$, is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass $m$ suspended by a string of length $l_2$, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $\dfrac{l_1}{l_2}$ is

A ball of mass 0.2 kg rests on a vertical post of height 5 m. A bullet of mass 0.01 kg, traveling with a velocity $V$ m/s in a horizontal direction, hits the centre of the ball. After the collision, the ball and bullet travel independently. The ball hits the ground at a distance of 20 m and the bullet at a distance of 100 m from the foot of the post. The initial velocity $V$ of the bullet is

A thin ring of mass 2 kg and radius 0.5 m is rolling without slipping on a horizontal plane with velocity 1 m/s. A small ball of mass 0.1 kg, moving with velocity 20 m/s in the opposite direction, hits the ring at a height of 0.75 m and goes vertically up with velocity 10 m/s. Immediately after the collision

A point mass of 1 kg collides elastically with a stationary point mass of 5 kg. After their collision, the 1 kg mass reverses its direction and moves with a speed of 2 m s⁻¹. Which of the following statement(s) is (are) correct for the system of these two masses?

A binary star consists of two stars A (mass $2.2M_s$) and B (mass $11M_s$), where $M_s$ is the mass of the sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is

Look at the drawing given in the figure which has been drawn with ink of uniform line-thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is $m$. The mass of the ink used to draw the outer circle is $6m$. The coordinates of the centres of the different parts are: outer circle $(0,0)$, left inner circle $(-a,a)$, right inner circle $(a,a)$, vertical line $(0,0)$ and horizontal line $(0,-a)$. The $y$-coordinate of the centre of mass of the ink in this drawing is

Two small particles of equal masses start moving in opposite directions from a point A in a horizontal circular orbit. Their tangential velocities are $v$ and $2v$, respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at A, these two particles will again reach the point A?

If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that

Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. These have masses $m$, $2m$ and $m$, respectively. The object A moves towards B with a speed 9 m/s and makes an elastic collision with it. Thereafter, B makes completely inelastic collision with C. All motions occur on the same straight line. Find the final speed (in m/s) of the object C.

Two balls, having linear momenta $\vec p_1=p\hat i$ and $\vec p_2=-p\hat i$, undergo a collision in free space. There is no external force acting on the balls. Let $\vec p_1'$ and $\vec p_2'$ be their final momenta. The following option(s) is(are) NOT ALLOWED for any non-zero value of $p,a_1,a_2,b_1,b_2,c_1$ and $c_2$.

The speed of the block at point B immediately after it strikes the second incline is

If collision between the block and the incline is completely elastic, then the vertical (upward) component of the velocity of the block at point B, immediately after it strikes the second incline is

STATEMENT-1: In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision.

because

STATEMENT-2: In an elastic collision, the linear momentum of the system is conserved.

STATEMENT-1: If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant.

because

STATEMENT-2: The linear momentum of an isolated system remains constant.