Automobile drives into a brick wall

[bmoore509]

Homework Statement

An automobile of mass 3000 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 6 × 106 N/m, and is observed to compress a distance of 4.23 cm as the car is brought to rest. What was the initial speed of the automobile?
Answer in units of m/s.

Homework Equations

F=-kx
Ke=(1/2)mv^2
U=(1/2)kx^2
Kf+Uf=Ki+Ui
W=Kf-Ki+Uf-Ui

The Attempt at a Solution

I'm really not sure where to start. I tried doing KE=U but I know that's not right.

 

[Stonebridge]

The kinetic energy of the car is absorbed by the bumper.
Use the spring formula (F=kx) and spring constant given to calculate the energy "stored" in the bumper when compressed 4.23cm
Energy stored will be average force times distance compressed.
Equate this to the kinetic energy of the car before collision.

 

[bmoore509]

I got it. I had a way to do it but the answer never seemed right. I didn't think about the fact that I needed to convert the 4.23 cm.

I did:
F=-kx = -253800
W=Fd = -10735.74
W=-.5mv^2+.5kx^2
-10735.74=-.5(3000)v^2-.5(6X10^6)(0.0423^2)
V=1.89m/s

 

[Stonebridge]

Good.
It was just a case of putting ½Fx² equal to ½mv²
Kinetic energy lost by the car is "absorbed" by the bumper.

 

[rwisz]

I would approach this problem by first identifying the relevant equations and variables. In this case, we have the mass of the car (m = 3000 kg), the spring constant (k = 6 × 106 N/m), and the distance the spring compressed (x = 4.23 cm or 0.0423 m). We also know that the car came to rest, so the final kinetic energy (Kf) is 0.

Next, I would use the equation F = -kx to find the force applied to the car by the spring. Plugging in the values, we get:

F = -kx = -(6 × 106 N/m)(0.0423 m) = -25400 N

We can then use the equation W = Kf - Ki + Uf - Ui to find the initial kinetic energy of the car (Ki). Since the car is initially at rest, Ui = 0. Plugging in the values, we get:

W = Kf - Ki + Uf - Ui
0 = 0 - Ki + (1/2)kx^2 - 0
Ki = (1/2)kx^2 = (1/2)(6 × 106 N/m)(0.0423 m)^2 = 254 N

Finally, we can use the equation Ke = (1/2)mv^2 to solve for the initial velocity (v) of the car. Plugging in the values, we get:

Ke = (1/2)mv^2
254 N = (1/2)(3000 kg)v^2
v = √(2(254 N)/(3000 kg)) = 2.53 m/s

Therefore, the initial speed of the automobile was 2.53 m/s. This means that the car was traveling at a relatively low speed before it hit the brick wall, which is why the compression distance of the bumper was relatively small. This information can be useful for designing safer cars and understanding the impact of collisions.