Collision

A sand-filled box of mass $M$ is suspended vertically by a rope of length $L$, forming a ballistic pendulum. A bullet of mass $m$ is fired horizontally into the box, causing it to swing. The bullet becomes embedded in the box, and the combined system swings up to a maximum angle $\theta$ with the vertical.

A block A ($m_A = 2.0$ kg) moves on a smooth tabletop at $v_A = 10$ m/s. It approaches block B ($m_B = 5.0$ kg), which has a light spring ($k = 1120$ N/m) attached to its rear. Block B is moving in the same direction at $v_B = 3$ m/s. Block A collides with the spring.

As shown in the figure, pendulum ball A is released from rest at an angle $\alpha = 45^\circ$. At the bottom of its swing, it collides head-on with the identical, stationary ball B. After the collision, ball B swings up to a maximum angle of $\beta = 30^\circ$. The masses of the balls are equal.

An $\alpha$ particle collides with an initially stationary oxygen nucleus. After the collision, the $\alpha$ particle is scattered at an angle of $64^\circ$ with its initial path. The oxygen nucleus recoils at an angle of $51^\circ$ on the other side of the initial path. The mass of the oxygen nucleus is four times that of the $\alpha$ particle.

Two small spheres, A and B, of equal radius move towards each other along the same straight line on a smooth horizontal surface. The mass of sphere A is greater than the mass of sphere B ($m_A > m_B$), and their kinetic energies are equal before the collision.

There are 5 small blocks (1, 2, 3, 4, 5), each with mass $m$ and negligible size. They are placed at equal distances $L$ on an inclined plane with an angle $\theta=30^\circ$. The incline is smooth above block 2 and rough below it. The coefficient of friction (static and kinetic) for all blocks on the rough section is $\mu$. Initially, block 1 is held at rest, while blocks 2, 3, 4, and 5 are at rest on the incline. Block 1 is then released, slides down, and collides with block 2. Subsequent collisions occur with the other blocks. All collisions are perfectly inelastic.

As shown in Figure, three rigid balls with masses $m_1$, $m_2=2m_1$, and $m_3=2m_1$ are placed in a horizontal, narrow, rigid circular groove fixed on a table. The balls are initially at positions I, II, and III, which are equidistant from each other. Friction is negligible. Initially, balls $m_2$ and $m_3$ are at rest, while ball $m_1$ moves along the groove with an initial velocity $v_0 = R\pi/2$, where $R$ is the effective radius of the groove. All collisions between the balls are perfectly elastic.

Two artificial satellites, each with mass $m = 200$ kg, are in the same circular orbit at an altitude equal to the Earth's radius, $h=R$. They are moving in opposite directions and eventually collide. The collision is perfectly inelastic. Gravitational forces between the satellites and air resistance are negligible. Use Earth's radius $R = 6.4 \times 10^6$ m and surface gravity $g = 10$ m/s$^2$.