The required spring constant:
The spring constant of the spring is [tex]12\times 10^4 \text{ N/m}[/tex].
Calculation:
The mass of the car is m=1200 kg, the speed of the car is v=25 m/s, and after colliding the spring is brought to rest at a distance of x=2.5m. Let the spring constant of the spring is, k.
From the conservation of energy,
Total initial kinetic energy= Total final potential energy of the spring
Therefore,
[tex]\frac{1}{2}mv^2=\frac{1}{2}kx^2[/tex]
Now, substituting the values of the mass of the car, speed of the car, and displacement, we get:
[tex]$\begin{aligned}1200\times(25)^2&=k\times (2.5)^2\\\Rightarrow k&=\frac{1200\times(25)^2}{(2.5)^2}\\&=12\times 10^4 \text{ N/m}\end{aligned}$[/tex]
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