Work-Energy Theorum: Spring potential energy vs Kinetic Energy

1. The problem statement, all variables and given/known data

A 1350-kg car rolling on a horizontal surface has a speed v = 40 km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.5 m. What is the spring constant of the spring? Ignore Friction and assume spring is mass-less.

2. Relevant equations
$$W = \Delta E$$
$$E_{pspring} = \frac{1}{2}(kx^2)$$
$$E_k = \frac{1}{2}(mv^2)$$

3. The attempt at a solution

First right off the bat, i converted 40 km/h to its m/s equivalent of aprox. 11.11 m/s

i state the law of conservation of energy: Energy before = Energy after

Therefore:

$$E_k = E_{pspring} \frac{1}{2}(mv^2) = \frac{1}{2}(kx^2)$$

then i isolate k

$$k = \frac{-mv^2}{x^2}$$

now heres the issue, is x negative? because the displacement is against the direction of motion?
and 2.5m = x, (-2.5)^2 gives me a answer of 4266 Nm
but -(2.5)^2 is entirely different.. This has been a long lasting math issue for me.

And what if x is positive?

i know k MUST be positive right?
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 Recognitions: Homework Help (2.5)^2 is correct. There is no negative energy in the nature.
 Recognitions: Homework Help There is no minus sign in mv^2 = kx^2 or in k = mv^2/x^2. No way you can get k negative! The minus sign in F = -kx is supposed to help keep track of the fact that the force of the spring is opposite to the direction of stretch but it does seem to have a habit of getting in the way. k is ALWAYS positive.

Work-Energy Theorum: Spring potential energy vs Kinetic Energy

Thanks, the negative sign on mv^2 was an algebra error... Thanks for the clarification guys!
 Your attempt is correct but you missed somethingthat for a spring if you take natural length as the datum, the force on change in length is given as: $$\vec{F}= -k \vec{x}$$ and hence work done by a spring against external forces $$W_{s}=\int\vec{F}\vec{.dx}$$ over the required limits in our case the answer is $$W_{s}=-\frac{kx^{2}}{2}$$ as $$W_{s}=\Delta E$$ $$\Delta E=-\frac{mv^{2}}{2}$$ the change part was where you lost it all...the KE FELL TO ZERO. HENCE A NEGATIVE CHANGE.

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