Proton Acceleration: Exploring Relative Speed

[Andrew Aidan]

This concept may have been discussed before, but I couldn't really find any that asked the same question. Through multiple sources, I have discovered that, in the LHC, we have the ability to propel protons to 99.99999% the speed of light (relative to a stationary object), and make them collide. But, what if we make them pass each other? Wouldn't one proton be traveling at 199.99998% the speed of light relative to the other? I asked one of my teachers about this and she said that it doesn't work that way, but couldn't explain why. Isn't that the definition of relativity? I'm a high school student working on a senior project (not as homework), so don't be afraid to simplify it (I actually encourage that).

 

[jtbell]

Wouldn't one proton be traveling at 199.99998% the speed of light relative to the other?

No, not by the customary definition of "speed of one object relative to another."

According to us, "standing still" in the laboratory, the distance between the two protons does indeed decrease (or increase, depending on whether they're approaching or receding) at 199.99998% of c. This speed is sometimes called "separation velocity," and it's perfectly OK for it to be greater than c.

The "speed of one proton relative to the other" is what we would observe if we were moving alongside one proton so that it was stationary relative to us. The speed of the other proton from that point of view is given by the relativistic "velocity addition" formula:

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/einvel.html

The result is always less than c.

 

[Andrew Aidan]

The "speed of one proton relative to the other" is what we would observe if we were moving alongside one proton so that it was stationary relative to us. The speed of the other proton from that point of view is given by the relativistic "velocity addition" formula

But if I were to substitute in the velocities of both particles (relative to a stationary object), then wouldn't their velocity (relative to each other), if they were moving at equal speeds, always equal zero, or did I plug something in wrong? I plugged in 0.9999999 for both v_A and v_B.

EDIT: For the relativistic projectile equation, I plugged in 0.6 for v_A, 0.99999 for v_p, and 0.99999 for v_B, and it came out to be 0.5999999. This would suggest that the projectile is moving at a slower rate for the incoming ship (B) than it is for the observing ship (A), but that doesn't make any logical sense.

 

[elfmotat]

You're plugging it in wrong.

 

[Andrew Aidan]

Care to explain what I've done wrong? Simply saying that I've done something wrong without explaining what's wrong doesn't help too much, seeing as how I plugged in numerous values in several different orders.

EDIT: I plugged in the value -0.99999 for v_B, and it gave me a more reliable answer of -0.9999999999499994%, but I still don't understand the latter situation:

For the relativistic projectile equation, I plugged in 0.6 for vA, 0.99999 for vp, and 0.99999 for vB, and it came out to be 0.5999999. This would suggest that the projectile is moving at a slower rate for the incoming ship (B) than it is for the observing ship (A), but that doesn't make any logical sense.

 

[mmzaj]

$$u\oplus v = \frac{v+u}{1+\frac{uv}{c^2}}$$

that's the relation you should be using . notice that the two protons are moving in opposite directions, hence the velocities are to be added.

 

[Andrew Aidan]

$$u\oplus v = \frac{v+u}{1+\frac{uv}{c^2}}$$

that's the relation you should be using . notice that the two protons are moving in opposite directions, hence the velocities are to be added.

Right, and that's what I used. The answer to my original question came out to be something that seems more realistic, but I still don't understand why they aren't just added.

Say a car, driving at a constant speed of exactly thirty MPH, passes a jogger running at exactly five MPH, in the opposite direction. To a stationary observer, the car is moving at 30 MPH, and the jogger 5 MPH. But, relative to the jogger, isn't the car driving by at 35 MPH and vice versa, or is this a different situation altogether? Am I thinking about this wrong?

 

[PAllen]

Right, and that's what I used. The answer to my original question came out to be something that seems more realistic, but I still don't understand why they aren't just added.

Say a car, driving at a constant speed of exactly thirty MPH, passes a jogger running at exactly five MPH, in the opposite direction. To a stationary observer, the car is moving at 30 MPH, and the jogger 5 MPH. But, relative to the jogger, isn't the car driving by at 35 MPH and vice versa, or is this a different situation altogether? Am I thinking about this wrong?

It's the same formula. Apply it and you can't detect the uv/c^2 term. However, for accelerator velocities, this term is large.

Mathematically, the relative to the jogger the car is not going 35 MPH, instead it is (approx):

34.99999999999999 (14 nines, if I got it right)

 

[nitsuj]

an initial "hurdle" with SR is that length & time are intuitively thought to be absolute measurements, in SR they are not. Since those two measurements compute a speed like c, it becomes particularly difficult to "see" the invariance of c (and the addition of velocities).

from a physics perspective, it is c that is "absolute", while measurements of length and time vary, that is measurements of length and time are not invariant. Said differently again, two people with a relative speed of say 35 mph will not measure time and length the same as each other.

As PAllen pointed out, it's not a different situation in the scenario you described, pretty cool I think.

The only thing I can think of that may help you perceive this more intuitively is to consider that the measurements of length and time will vary depending on the relative velocity of the "person" making the measurements.

A "frame of Reference" (often typed as FoR) is a term that is typically used for distinguishing such "persons" making measurements.

In the scenario you proposed above, there are three Frames of Reference; the jogger, the car and the "stationary" observer (ground speed 0).

As PAllen calculated, the jogger would calculate the speed of the car to be 34.99999999999999mph.

I got a question too now,

PAllen, is that calculated value what the jogger would calculate? Or is it what both would calculate (with different values for time/length "proportions")

 

[PAllen]

To someone standing on the road, the closing speed of car and jogger is 35 mph. Closing speed (how fast you observe the distance between to objects shrinking) can reach 2c. The speed of car as measured by jogger = speed of jogger as measured by car = 34.99999999999999 (appx.).

 

[mmzaj]

i think the best way for you to grasp the "counter intuitive" nature of SR is to try and understand lorentz transformations as rotations in 4 dimensions. the analogy between these 4 dimensional rotations and the classical 3 dimensional rotations makes the math a bit more intuitive. every prediction by SR is due to lorentz transformations. furthermore, it makes a good intro to GR.

 

[nitsuj]

To someone standing on the road, the closing speed of car and jogger is 35 mph. Closing speed (how fast you observe the distance between to objects shrinking) can reach 2c. The speed of car as measured by jogger = speed of jogger as measured by car = 34.99999999999999 (appx.).

Ah kk, that was my guess. Because the speed calculated by the jogger & car reminded me of intervals.

EDIT: lol it is an interval..doh! I never put the two together what a "keystone" that was. :) speed=interval

 

[Andrew Aidan]

Ok, this is making a bit of sense now. Thanks, guys,