Car hits brick wall, find inital speed of car

[AHinkle]

Homework Statement

An automobile of mass 2000 kg is driven into
a brick wall in a safety test. The bumper
behaves like a Hooke’s-law spring. It has an
effective spring constant of 5 × 106 N/m, and
is observed to compress a distance of 1.67 cm
as the car is brought to rest.
What was the initial speed of the automo-
bile?
Answer in units of m/s.

Homework Equations

[tex]\Delta[/tex]KE = (1/2)mv_f²-(1/2)mv_i²
W_{applied to spring}=(1/2)kx_f²-(1/2)kx_i²

The Attempt at a Solution

[tex]\Delta[/tex]x = -0.167m

W_app=(1/2)kx_f²-(1/2)kx_i²
W_app=(1/2)(5x10⁶)(-0.167)²-(1/2)(5x10⁶)(0)²

W_app = 69722.5 J

V_f = 0 (because the car is brought to rest)
m = 2000 Kg

[tex]\Delta[/tex]KE = 69722.5 J =(1/2)(2000)(0)²-(1/2)(2000)v_i²

69722.5 J = -(1000)v_i²
v_i² = (69722.5/-1000)

v_i = (-69722.5/1000)^1/2

and here i came to a dead end with a non-real answer. Err..
am I at least going in the right direction. I must have my sign off somewhere. I just want to know if I'm approaching the problem correctly.

 

[Stonebridge]

The change in the kinetic energy of the car, the energy it loses, is transferred to the buckle.
The car goes from v to zero so its change in kinetic energy is simply ½mv²
If the bumper behaves as a Hooke's Law spring, then the energy absorbed by it is given by a formula involving k, the spring constant, and x the distance it compresses.
You have all the information [m,k and x] to calculate v.
Do you know the spring formula?

 

[AHinkle]

do you mean
F_spring=-kx

and the work done by a spring is
W_s=[tex]\int[/tex](-kx) dx = 1/2kx_f²-1/2kx_i²

 

[Stonebridge]

do you mean
F_spring=-kx

and the work done by a spring is
W_s=[tex]\int[/tex](-kx) dx = 1/2kx_f²-1/2kx_i²

That's it. It's just ½kx² here where x is the compression. Equate the work done to the loss in k.e. of the car.

 

[AHinkle]

1/2kx_f²-1/2kx_i²=1/2mv_i²

1/2(5x10⁶)(-.0167)²-1/2(5x10⁶)(0)²=1/2(2000)v_i²

v_i = .835m/s? aww crap... i got a sign or something mixed up somewhere or i just don't understand what orientation i need to put all the parts together in

 

[AHinkle]

actually i checked the answer and .8350 m/s is correct! I found out what i did wrong. I converted from centimeters to meters incorrectly in my first work calculation and that threw everything off.

 

[Stonebridge]

Yes, the answer is fine. :)