Original link: https://www.yinwang.org/blog-cn/2022/10/09/relativity
Relativity Derivation Exercise
Below are a few exercises I have devised for deriving the theory of relativity. If you do the first three, you can derive the correct formula for relativity.
These exercises have been tested by several liberal arts students, science students, and a doctor. All of these people can independently derive and understand the theory of relativity in under an hour, all you need is a little elementary school math and physics.
basic exercises
Exercise 1
Suppose now that the world has changed and the speed of light is only 1 m/s (meters per second). That is, something that happens n meters away (such as a cat there) won’t be visible to you for n seconds. Now there is a cat that runs away from you at a constant speed of 0.5 m/s. If you observe the speed of the cat, you will see how fast the cat is?
Note : This 0.5 m/s is the speed felt by the cat, and the speed you measure may be different. How does a cat know its own speed? You can assume that there are scales drawn on the ground in advance, and the cat can know the distance from you by looking at these scales. It has a watch on it and can know the time. How do you measure a cat’s speed? You also look at the scales on the ground, and you also have a watch in your hand.
(You can start doing exercise 1 now. If you get stuck, click here again for tip 1. I’ll provide a link to the tip in a day.)
Thinking 1 : According to Hint 1, the “average speed” in the 1 m distance is calculated, and what we need is the “instantaneous speed”. Please design a new scheme by yourself to calculate the “instantaneous speed” of the cat in multiple positions. Are they all the same? Does this mean that the movements of the cat you observed were also uniform? (If you can’t do this, you can skip it first and continue with exercise 2.)
Exercise 2
As in exercise 1, the speed of light is still only 1 m/s, but instead of leaving you this time, the cat is running towards you at a speed of 0.5 m/s. What is the cat’s speed measured this time?
Thinking 2 : Please think, why do you observe different speeds when the cat leaves you and when it approaches you? Please imagine a situation in everyday life and use it to understand this matter.
Exercise 3
According to the ideas of the above two exercises, replace the specific numbers with variables, and derive a general formula. The cat leaves you at the speed of va, and the speed of light is c. Please measure the relationship between the cat’s speed (v s ) and va a . The result should be of the form v s = … , where v a is included in the expression on the right.
Definition of variables:
v a : the speed felt by the cat itself
v s : the speed of the cat you observed
c: speed of light
Verification formula : Substitute the specific numbers from Exercise 1 and Exercise 2 into the formula derived from Exercise 3 to see if it matches the result calculated before? Note that v a in the formula means “the speed of the cat’s departure”, if the cat is running towards you, you can use a negative number to represent v a .
So far, you have deduced the correct relativistic velocity transformation formula. If you have studied Einstein’s formula of relativity, you will find that the formula we derive here is different from Einstein’s formula. Einstein’s formula is wrong. If you want to understand where he went wrong, continue with the following exercise.
advanced practice
The following are some advanced thinking and extended exercises, all of which only require junior high school physics knowledge to solve, but some may require more thinking, and can be skipped if they cannot be done.
Exercise 4
Assuming that the cat’s seen length (actual length) when stationary is 1 meter, calculate:
a) What length do you observe when it leaves you at 0.5 m/s?
b) What length do you observe when it approaches you at 0.5 m/s?
Using the ideas from (a) and (b), derive a conversion relationship between the observed object length ls and the object ‘s actual length la.
Exercise 5
Please find a way to derive it yourself (what you see):
- Time conversion formula for the watch on the cat
- cat mass (mass) conversion formula
- Cat’s kinetic energy conversion formula
- Momentum conversion formula for cats
- cat color (frequency of light) conversion formula
It’s a bit difficult here, you can skip it if you can’t do it.
Exercise 6
Referencing Einstein’s formula of special relativity is different from the formula we deduced. How would the results from his formula differ for exercise 1 and exercise 2? Do you think there is anything wrong with Einstein’s formula? Whose formula is correct?
Exercise 7
Understanding the ” twin paradox ” of relativity, why does it happen? Would twin paradox still happen if we used the relativistic formula we derived here?
Exercise 8 (difficult)
This exercise is a little harder. Refer to Einstein’s E = mc 2 paper ( link to the paper) to understand how he derived the “mass-energy equation” E = mc 2 . Can you see what’s wrong with his derivation? What should the correct “mass-energy equation” look like?
That’s all for the exercises, and I’ll provide some hints and answers later.
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