Astronomers detect highest energy cosmic ray since 1991

Note that in modern terminology, the term "rest mass" is no longer preferred. It is now considered cleaner to use the term mass as a constant. The greater difficulty of accelerating as you get closer to the speed of light is then just a different formula for Newton's second law, instead of a change in the mass ("relativistic mass").

This is all equivalent mathematically, but it seems cleaner in terms of terminology rather than saying that the mass of an object depends on the relative speed to the observer.

> A proton moving at 99.5% the speed of light is not moving much faster than one moving at 99%, but the former has much more energy

If those were spaceships instead of protons one might wonder, given the large energy difference, if it would be worth spending the energy to go at 99.5% c instead of 99% c.

For a 4.3 lightyear trip, the 99% c ship would take 4.34 years. The 99.5% ship would take 4.32 years. That's only 8 days faster, which at first doesn't seem like it could be worth it. It only knocks about 0.5% off the trip.

But that's from the frame of reference of Earth. From the frame of reference of the ships the time will be shortened due to relativistic time dilation.

The 99% c ship makes the trip in 0.613 years (224 days), and the 99.5% c ship makes the trip in 0.432 years (156 days). That's 30% knocked off the trip--which would mean a substantial savings in supplies for the crew. That might be enough to make the extra energy to get that extra 0.5% speed worth it.

If anyone wants to play around with these kind of things I have a calculator designed to make these kind of special relativity calculations fairly easy. Source here [1], deployed here [2].

Here's how you would use it to get the numbers show above. Enter 299792458 in the "scale" field (which makes it so that the distances are in lightyears and time in years).

Enter the speed of the ship in the velocity field (0.99 or 0.995).

The fields in the calculation rows represent an x and a t in your frame (fields 1 and 2) and an x and a t in the ship's frame (fields 3 and 4).

At any given time two fields are input fields (labeled with ① and ②) and the other two fields are outputs. Enter values in the inputs and it sets the outputs such that the event at the x, t in your frame given by fields 1 and 2 is the same as the event at the x, t in the ship's frame given by fields 3 and 5. (It is assumed that at our x = 0, t = 0 and the ship's x = 0, t = 0 are the same).

You can change which fields are inputs by clicking on the symbol to the left of a field. That will make that field input ①, and the previous ① will become ②.

For finding out how long the 4.3 light year trip takes in the ship's frame, we are looking for when the point that is at x = 4.3 in our frame is at x = 0 in the ship's frame.

So make fields 1 and 3 (our x and ship's x) inputs, and put 4.3 in our x and 0 in the ship's x. Then field 2 (our t) shows when the ship arrives according to our clock, and field 4 (ship's t) shows what time is on the ship's clock.

[1] https://github.com/tzs/Physics-special-relativity-calculator

[2] https://tzs.github.io/Physics-special-relativity-calculator/