Can Spring Physics Predict Collision Outcomes?

[etan]

This was a problem on my final exam that I couldn't figure out, it drove me insane. Someone please tell me it's impossible

A spring has a 1kg block attached to it and a k of 30,000. A 2kg block moving at 30m/s over a frictionless surface hits and sticks to the 1kg block. What is the displacement of the spring? What frequency will it vibrate after impact?

I figured out that the final velocity of both blocks will be 20m/s and the final mass will be 3kg. I could not figure out the final acceleration to use m*a=x*k

I was not given the distance between the blocks or the time.

Any ideas?

 

[chroot]

Step 1) What is the kinetic energy of the resulting 3 kg lump?

Step 2) The work done by the lump in compressing the spring can be found by

$$W = \int F\, dx<br /> = \int kx\, dx<br /> = \frac{1}{2} kx^2$$

where W is equal to the kinetic energy of the lump.

The angular velocity of oscillation of a system with "springiness" k and mass m is

$$\omega = \sqrt{\frac{k}{m}$$

The frequency of an oscillation with angular velocity $\omega$ is

$$f = \frac{\omega}{2 \pi}$$

Does this make sense?

- Warren

 

[M98Ranger]

Hi there,

First of all, don't worry too much about not being able to solve this problem on your final exam. Physics can be challenging and it's okay to struggle with certain concepts or problems.

To solve this problem, you will need to use the conservation of momentum and the conservation of energy equations. The conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. In this case, the momentum before the collision is 2kg * 30m/s = 60kg*m/s. After the collision, the total mass is 3kg and the final velocity is 20m/s, so the total momentum is 3kg * 20m/s = 60kg*m/s. This means that the momentum is conserved.

Next, we can use the conservation of energy to find the displacement of the spring. The total energy before the collision is kinetic energy, which is equal to (1/2)mv^2 = (1/2)(2kg)(30m/s)^2 = 900J. After the collision, the total energy is potential energy stored in the spring, which is equal to (1/2)kx^2. So, we can set these two equations equal to each other and solve for x:

900J = (1/2)(30,000)(x^2)
x^2 = 900J / 15,000
x = √0.06 = 0.24m

Therefore, the displacement of the spring is 0.24m.

To find the frequency of vibration, we can use the equation f = (1/2π)√(k/m). In this case, the mass is 3kg (1kg + 2kg) and the k value is 30,000. So, the frequency would be:

f = (1/2π)√(30,000/3) = 61.24Hz

I hope this helps and good luck with your future physics studies! Remember, it's always okay to ask for help when you're struggling with a problem. Keep practicing and you'll get the hang of it.