An engine is attached to a wagon through a shock absorber of length 1.5 m . The system with a total mass of $40,000 \mathrm{~kg}$ is moving with a speed of $72 \mathrm{kmh}^{-1}$ 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, the spring constant is
$\_\_\_\_$ $\times 10^5 \mathrm{~N} / \mathrm{m}$.
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Solution
$$
\begin{aligned}
&\text { Vork }=\Delta K . E .\\
&\begin{aligned}
& W_{\text {tration }}+W_{\text {sping }}=0-\frac{1}{2} m v^2 \\
& -\frac{90}{100}\left(\frac{1}{2} m v^2\right)+W_{\text {apmg }}=-\frac{1}{2} m v^2 \\
& W_{\text {sumg }}=-\frac{10}{100} \times \frac{1}{2} m v^2 \\
& -\frac{1}{2} k x^2=-\frac{1}{20} m v^2 \\
& \Rightarrow \quad k=\frac{40000 \times(20)^2}{10 \times(1)^2}=16 \times 10^5
\end{aligned}
\end{aligned}
$$
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