Elastic Potential Energy and Hooke's Law Problem

[y201]

Homework Statement

A small truck is equipped with a rear bumper that has a spring constant of 800,000 N/m. the bumper can be compressed up to 15cm without causing damage to the truck. What is the maximum velocity with which a solid 1000-kg car can collide with the bumper without causing damage to the truck?

Homework Equations

Kinetic Energy=(1/2)(m)(v^2)
Elastic Energy=(1/2)(k)(x^2)

The Attempt at a Solution

I tried to make those two equations equal to one another and then isolate for "v" and solve for it. however, i got the wrong answer. the answer at the back of the book is 34 m/s. this problem is based on elastic potential energy and hooke's law. all help is appreciated.

 

[Birkeland]

Did you convert the 15cm to 0.15m? Otherwise the process you described is correct.

Actually, I just did the problem backwards. Using their "correct answer" of 34 m/s means the bumper must move 1.5 m, not 0.15. Basically the back of you book is wrong.

 

[y201]

thanks a lot birkeland for your help.
what did you get as the right answer?

 

[y201]

Did you convert the 15cm to 0.15m? Otherwise the process you described is correct.

Actually, I just did the problem backwards. Using their "correct answer" of 34 m/s means the bumper must move 1.5 m, not 0.15. Basically the back of you book is wrong.

what answer did you get using the 0.15m?

 

[rwisz]

To solve this problem, we can use the conservation of energy principle, which states that the total energy of a system remains constant. In this case, the initial kinetic energy of the car will be converted into elastic potential energy of the bumper at maximum compression, and then back into kinetic energy as the bumper pushes the car back.

We can set up the equation as follows:

Kinetic Energy of car = Elastic Potential Energy of bumper

(1/2)(m)(v^2) = (1/2)(k)(x^2)

Where m is the mass of the car (1000 kg), v is the maximum velocity of the car, k is the spring constant of the bumper (800,000 N/m), and x is the maximum compression of the bumper (15 cm = 0.15 m).

Solving for v, we get:

v = √[(kx^2)/m]

Substituting the values, we get:

v = √[(800,000 N/m)(0.15 m)^2 / 1000 kg] = 34 m/s

Therefore, the maximum velocity with which the car can collide with the bumper without causing damage to the truck is 34 m/s. This means that if the car is travelling at a speed higher than 34 m/s, it will cause damage to the truck's bumper.