Collision

As shown in the figure, a steel ball of mass $m$ is shot horizontally with velocity $v$ into the barrel of a spring gun of mass $M$, which is initially at rest on a smooth horizontal surface. The friction between the spring and the barrel is negligible.

Two small balls of equal mass move towards each other at the same speed. After a head-on collision, they both move in the opposite direction at their original speeds.

Two objects, A and B, move in the same direction on the same straight line, with A in front. Object A has a mass of $m_A = 2$ kg and a speed of $v_A = 1$ m/s. Object B has a mass of $m_B = 4$ kg and a speed of $v_B = 3$ m/s. After B catches up to A for a head-on collision, both objects continue in the original direction. The speed of A becomes $v_A' = 3$ m/s, and the speed of B becomes $v_B' = 2$ m/s.

Consider a collision between two objects with masses $m_1$ and $m_2$. Their velocity changes are defined as $\Delta v_1 = v_1' - v_1$ and $\Delta v_2 = v_2' - v_2$, where $v$ and $v'$ are the velocities before and after the collision, respectively.

A proton (mass $m_p = 1.67 \times 10^{-27}$ kg) moving at $v_p = 1.0 \times 10^7$ m/s collides with a nitrogen nucleus initially at rest ($v_N = 0$). After the collision, the proton rebounds at $v_p' = -6.0 \times 10^6$ m/s, and the nitrogen nucleus moves forward at $v_N' = 4.0 \times 10^6$ m/s. The velocities are along a straight line.

An object with mass $m_1=2.0$ kg undergoes a one-dimensional elastic collision with a stationary object of mass $m_2$. After the collision, the first object continues to move in its original direction with a speed equal to one-fourth of its initial speed.

As shown in Figure, a bullet with mass $m = 50$ g is fired into a wooden block of mass $M=500$ g. The block is attached to a spring with spring constant $k=1000$ N/m. The bullet embeds itself in the block, and the combination compresses the spring by a maximum distance of $x=50$ cm. The horizontal surface is frictionless.

A coal car moves at a constant speed of $v=3$ m/s under a hopper. Coal is dropped vertically into the car at a rate of $\lambda=500$ kg/s. Ignore friction.

There are two small balls, A and B, of equal mass. Ball A collides with ball B, which is initially at rest on a smooth horizontal table. The initial velocity of ball A is $v_1 = 30$ m/s to the right. After the collision, ball A moves in a direction at an angle $\alpha = 30^\circ$ with its original direction of motion. Ball B's velocity makes an angle $\beta = 45^\circ$ with the original direction of motion of ball A.

Two balls move towards each other, each with a speed of 2 m/s. One ball's mass is 3 times the other's. They stick together after colliding.