Spring Car, Acceleration Problem: Find Spring Constant

[Jaccobtw]

Homework Statement

You devise a wound up car powered by a spring for trips to the grocery store. The car has an inertia of 500 kg and is 4.2 m long. It should be able to accelerate from rest to 20 m/s at least 50 times before the spring needs winding. The spring runs the length of the car, and a full winding compresses it to half of its length In order to meet the acceleration requirement, what must the value of the spring constant be?

Relevant Equations

F = ma
F = -kd

"It should be able to accelerate from rest to 20 m/s at least 50 times before the spring needs winding"
-So F = -kd = -k(2.1) - d is 2.1 because it is the compression length

Now, since we know the d, divide it by 50, 2.1/50 = 0.042m

Basically, the spring unwinds 0.042 m 50 times for a total distance of 2.1m.

Now, calculate acceleration:
v(f)^2 = v(i)^2 * 2 * a * d

We know our final velocity is 20 m/s, our initial is 0.

Solving for a we get, 4,762 m/s^2

Back to our original equations

F = -kd = ma

Solve for k and we get : 5.7 * 10^7 N/m

But I got this wrong :(

 

[haruspex]

the spring unwinds 0.042 m 50 times

No. The energy stored in a spring rises as the square of the displacement, so the first 0.042 m it unwinds releases more energy than the next 0.042 m, etc.
Instead of forces, work with energy.