An engine is attached to a wagon through a shock absorber of length $1.5\,m$. The system with a total mass of $50,000 \,kg$ is moving with a speed of $36\, km\,h^{-1}$ when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by $1.0\,m$.

If $90\%$ of energy of the wagon is lost due to friction, calculate the spring constant.

To simulate car accidents, auto manufacturers study the collisions of moving cars with mounted springs of different spring constants. Consider a typical simulation with a car of mass $1000\; kg$ moving with a speed $18.0\; km / h$ on a rough road having $\mu$ to be $0.5$ and colliding with a horizontally mounted spring of spring constant $6.25 \times 10^{3} \;N m ^{-1} .$ What is the maximum compression of the spring in $m$?

A ball of mass $2 \,m$ and a system of two balls with equal masses $m$ connected by a massless spring, are placed on a smooth horizontal surface (see figure below). Initially, the ball of mass $2 \,m$ moves along the line passing through the centres of all the balls and the spring, whereas the system of two balls is at rest. Assuming that the collision between the individual balls is perfectly elastic, the ratio of vibrational energy stored in the system of two connected balls to the initial kinetic energy of the ball of mass $2 \,m$ is

A toy gun fires a plastic pellet with a mass of $0.5\  g$. The pellet is propelled by a spring with a spring constant of $1.25\  N/cm$, which is compressed $2.0\  cm$ before firing. The plastic pellet travels horizontally $10\  cm$ down the barrel (from its compressed position) with a constant friction force of $0.0475\  N$. What is the speed (in $SI\  units$) of the bullet as it emerges from the barrel?

The length of a spring is a when $\alpha $ force of $4\,N$ is applied on it and the length is $\beta $ when $5\,N$ force is applied. Then the length of spring when $9\,N$ force is applied is