A railroad car of mass $M$ is at rest on frictionless rails when a man of mass $m$ starts moving on the car towards the engine. If the car recoils with a speed $v$ backward on the rails, with what velocity is the man approaching the engine?

Two persons each of mass $m$ are standing at the two extremes of a railroad car of mass $M$ resting on a smooth track. The person on left jumps to the left with a horizontal speed $u$ with respect to the state of the car before the jump. Thereafter, the other person jumps to the right, again with the same horizontal speed $u$ with respect to the state of the car before his jump. Find the velocity of the car after both the persons have jumped off.

Figure shows a small block of mass $m$ which is started with a speed $v$ on the horizontal part of the bigger block of mass $M$ placed on a horizontal floor. The curved part of the surface shown is semicircular. All the surfaces are frictionless. Find the speed of the bigger block when the smaller block reaches the point $A$ of the surface.

Consider a head-on collision between two particles of masses $m_1$ and $m_2.$ The initial speeds of the particles are $u_1$ and $u_2$ in the same direction. The collision starts at $t=0$ and the particles interact for a time interval $\mathrm{\Delta }t$. During the collision, the speed of the first particle varies as $v\left(t\right)=u_1+\frac{t}{\mathrm{\Delta }t}\left(v_1-u_1\right)$.

Find the speed of the second particle as a function of time during the collision.