Solving the T and U Paradox: A Urgent Homework Challenge

[KiyoEtAlice]

URGENT! T and U paradox

Homework Statement

I'm really confused about how I should do this problem and this is urgent so if anyone can help, it would be very much appreaciated. Thanks.

T and U Bar paradox (This is my assigned homework, it is an edit of question 6-5 from a textbook called Spacetime Physics, it's one of my teacher's collection so I don't know anything more):

A U shaped structure made of the strongest steel contains a detonator switch connected by wire to one metric ton of explosive TNT. A T-shaped structure made of the same strong steel fits inside U, with the long arm of T not quite long enough to reach the detonator switch when the two collide. Therefore there will be no explosion.

But, look at the same situation in the rest frame of the T structure . In this frame the arm of the T has its rest length, while the two arms of the U-structure are Lorentz-contracted. Therefore the arm of the T will certainly strike the detonator switch and there will be an awful explosion.

Q: ==> Make two spacetime diagrams (the bigger the better) representing the motion of the objects in the T frame and in the U frame. Take the relative speech to be 87% of the speed of light --then the contraction is half of the rest length. Label the front and back of the T structure P and Q respectively and A and B the front and back of the U structure. Determine from your diagram whether there will be an explosion or not. And explain Why your diagram gives your answer.

Homework Equations

**I've attached the image of this paradox that I found on the net to make things clear (and yes the image is right). Please help me out on the drawings and explanation, I'm not really sure how to do it. Note: the diagram is not the solution, it is just another version of the diagram that goes with the question.

The Attempt at a Solution

Well, I am completely blank at this. I know how it should be like, and I know that it wouldn't explode.

Thanks!

 

[anathema]

Dear student,

Firstly, let's start by understanding the setup of the problem. In the U frame, we have a U-shaped structure made of strong steel, with a detonator switch connected to one metric ton of explosive TNT. Inside the U structure, we have a T-shaped structure made of the same strong steel, with one arm of the T being slightly shorter than the other arm. This means that when the two structures collide, the shorter arm of the T will not reach the detonator switch, preventing an explosion.

However, when we look at the same situation in the rest frame of the T structure, things appear differently. In this frame, the T structure is at rest, with its arms having their proper lengths. However, the U structure is moving at a speed of 87% of the speed of light, causing its arms to be Lorentz-contracted. This means that in the T frame, the U structure appears to be shorter, and the shorter arm of the T structure could potentially reach the detonator switch, causing an explosion.

To better understand this paradox, we need to create two spacetime diagrams, one for the U frame and one for the T frame. In these diagrams, we will plot the motion of the two structures, with time on the vertical axis and distance on the horizontal axis.

Let's start with the U frame. Since the U structure is at rest in this frame, its spacetime diagram will be a straight vertical line, representing its constant position in space. The T structure, on the other hand, will be moving towards the U structure with a speed of 87% of the speed of light. This means that its spacetime diagram will be a diagonal line, representing its constant motion towards the U structure. The point where the two lines intersect represents the moment of collision between the two structures.

Now, let's look at the T frame. In this frame, the T structure is at rest, so its spacetime diagram will be a straight vertical line, just like in the U frame. However, the U structure is now moving towards the T structure with a speed of 87% of the speed of light. This means that its spacetime diagram will be a diagonal line, just like the T structure in the U frame. The point where the two lines intersect represents the moment of collision between the two structures.

Now, let's analyze the situation. In the U frame, the collision occurs at a point where the

 

[baba89]

Dear student,

Thank you for reaching out for help with this homework challenge. The T and U paradox is a famous thought experiment that explores the concept of length contraction in special relativity. It is a challenging problem, but with some careful analysis, we can come to a solution.

First, let's clarify the situation. We have a U-shaped structure made of strong steel, with a detonator switch connected to a metric ton of explosive TNT. Inside the U structure, there is a T-shaped structure made of the same strong steel. The long arm of the T is not long enough to reach the detonator switch when the two structures collide, so there will be no explosion.

However, when we look at the same situation from the rest frame of the T structure, things appear differently. In this frame, the arm of the T has its rest length, while the two arms of the U structure are Lorentz-contracted. This means that the arm of the T will strike the detonator switch, causing an explosion.

To analyze this paradox, we need to draw spacetime diagrams for both frames of reference. In the T frame, the T structure is at rest, so its worldline is a straight vertical line. The U structure is moving at 87% of the speed of light, so its worldline will be tilted at an angle. Label the front and back of the T structure as P and Q, respectively, and the front and back of the U structure as A and B. The distance between P and Q represents the rest length of the T structure, while the distance between A and B represents the contracted length of the U structure.

In the U frame, the U structure is at rest, so its worldline is a straight vertical line. The T structure is moving at 87% of the speed of light, so its worldline will be tilted at an angle. Label the front and back of the T structure as P' and Q', respectively, and the front and back of the U structure as A' and B'. The distance between P' and Q' represents the length of the T structure as seen from the U frame, while the distance between A' and B' represents the rest length of the U structure.

From these spacetime diagrams, we can see that in the T frame, the T structure will not reach the detonator switch, so there will be no explosion. In the U frame, however, the T structure will reach the