# A seemingly simple yet paradoxical question? (springs & collision)

• lillybeans
In summary, the problem involves a 8g bullet colliding with a 450g block of wood, compressing a spring with a spring constant of 80N/m by 0.15m. The goal is to find the velocity of the block and bullet when they first begin moving together, as well as the initial kinetic energy of the bullet. The solution involves calculating the elastic potential energy of the spring, using the equation Ee=1/2kx^2, and setting it equal to the initial kinetic energy of the bullet, using the equation Ek=1/2mv^2. The minimum separation velocity is found using the conservation of momentum equation, and the initial kinetic energy of the bullet is found using the equation
lillybeans

## Homework Statement

A 8g bullet hits a block of wood with mass of 450g which is at rest. The impact compresses the spring by 0.15m. The spring constant is 80N/m.

a) Find the velocity of the block and bullet when they first begin to move together.
b) Find the initial kinetic energy of the bullet.

HERE IS THE PROBLEM. I did this question before (it's from a physics test), I got it right, but now I no longer understand the reasoning behind my solution. It makes absolutely no sense.

## The Attempt at a Solution

Here is what I did, and apparently, it's right. I just don't understand why.

a)Ee=1/2kx2=(1/2)(80)(0.15)2=0.9J

Ek=1/2mv2
0.9=1/2(0.458kg)v2
v=1.982m/s

b)m1v1+m2v2=m(1+2)*Vmin <--minimum separation
0.008v + 0 = 0.458*1.982
v1=113.47

Ek=1/2(0.008)(113.47)2
=51.5J

-------------------------------------

Here is why I no longer think my solution makes sense.

Total energy before collision = kinetic energy in the bullet
Total energy during collision (minimum separation)= all converted to stored energy in the spring
Total energy after collision = kinetic energy again

Problem 1: notice, during minimum separation, Et=Est, which means there IS NO kinetic energy! Which means, it's not possible to find the velocity at minimum separation since it's all stored energy and no kinetic energy. Thus, shouldn't the minimum separation velocity be zero, and not 1.982m/s?

Problem 2: If all the system's energy has been converted to stored energy in the spring during minimum separation, doesn't that mean Et=Est=0.9J? Since Et before and after collision does not change, doesn't it imply that the initial kinetic energy of the bullet is also 0.9J, and not 51.5J? Since Et=Ek initial?

I know it seems perfectly obvious to you, but what exactly do Et and Est stand for? And what exactly is this "minimum separation" of which you speak?

vela said:
I know it seems perfectly obvious to you, but what exactly do Et and Est stand for? And what exactly is this "minimum separation" of which you speak?

Yes Ma'am, My apologies!

Et= Total Energy
Est = Stored Energy
Minimum separation = The minimum distance between the two objects during collision. It is also the point where when the two objects can be thought of as a "complex" and treated as one ginormous .. ugh.. thing.

I think you're just misunderstanding what happens in the problem.

The time scale of the collision is very short, so the spring doesn't compress at all during the collision. The bullet flies in and collides with the block. The block/bullet combo begins moving. It compresses the spring, which eventually brings them to rest. So immediately after the collision, there is no energy stored in the spring. It's all kinetic energy of the block/bullet combo.

By the way, I'm a dude.

vela said:
I think you're just misunderstanding what happens in the problem.

The time scale of the collision is very short, so the spring doesn't compress at all during the collision. The bullet flies in and collides with the block. The block/bullet combo begins moving. It compresses the spring, which eventually brings them to rest. So immediately after the collision, there is no energy stored in the spring. It's all kinetic energy of the block/bullet combo.

By the way, I'm a dude.

Thank you for answering, but what does it mean when the question asks "find the velocity when the block and the bullet first begin moving together"? Is that when the bullet has JUST touched the block right before it compresses the spring? Or is it AFTER the spring has been compressed, right before the two are about to launch off/separate again?

Also, another question. Assuming part b)'s answer is correct, that means we started off with 51.5J of kinetic energy in the bullet, and during the spring compression phase, where all the kinetic energy is converted to stored energy, we only have 0.9J of energy in total? Where did the 50.6J of energy go? And I'm pretty sure this is an elastic collision (no energy is lost)?

P.S. I'm sorry. I thought Vela is a woman's name.

Last edited:
I believe that the implication is that the bullet becomes embedded in the block of wood. Thus the situation is that of an inelastic collision. Energy will be lost.

## 1. How can a seemingly simple spring and collision scenario have a paradoxical outcome?

A seemingly simple spring and collision scenario can have a paradoxical outcome because of the counterintuitive properties of springs and collisions. While we may expect that a spring will always stretch or compress in response to an external force, this is not always the case. Additionally, collisions can result in unexpected changes in velocity and momentum, leading to seemingly paradoxical outcomes.

## 2. Can you provide an example of a seemingly simple yet paradoxical question involving springs and collisions?

One example of a seemingly simple yet paradoxical question involving springs and collisions is the "elastic band paradox." This scenario involves two identical elastic bands, one stretched and one relaxed, being dropped onto a flat surface. While our intuition may suggest that the stretched band would bounce higher, both bands often end up bouncing to the same height, leading to a paradoxical outcome.

## 3. What factors contribute to the paradoxical outcomes of spring and collision scenarios?

The paradoxical outcomes of spring and collision scenarios are largely due to the complex interplay between energy, momentum, and forces. Springs have the ability to store and release potential energy, and collisions involve the transfer of kinetic energy and momentum. These factors, along with other variables such as the mass and elasticity of the objects involved, can lead to seemingly paradoxical results.

## 4. How do scientists study and explain paradoxical outcomes in spring and collision scenarios?

Scientists use mathematical models and experiments to study and explain paradoxical outcomes in spring and collision scenarios. By using principles of physics, such as conservation of energy and momentum, scientists can create models that accurately predict the behavior of springs and collisions. They also conduct experiments to test these models and gain a better understanding of the factors that contribute to paradoxical outcomes.

## 5. Can these paradoxical outcomes be applied in real-world situations?

Yes, the paradoxical outcomes observed in spring and collision scenarios can have real-world applications. For example, the "elastic band paradox" has been used to explain the behavior of rubber tires in drag racing. Additionally, understanding the principles behind these paradoxical outcomes can help engineers design more efficient and effective structures and machines.

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