Work-Energy Theorum: Spring potential energy vs Kinetic Energy

[Senjai]

[SOLVED] Work-Energy Theorum: Spring potential energy vs Kinetic Energy

Homework Statement

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.

Homework Equations

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

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:

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

then i isolate k

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

now here's 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?

 

[rl.bhat]

(2.5)^2 is correct. There is no negative energy in the nature.

 

[Delphi51]

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.

 

[Senjai]

Thanks, the negative sign on mv^2 was an algebra error... Thanks for the clarification guys!

 

[vaibhav1803]

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.