Imagine you are watching a person travel at 99% the speed of light in a vacuum (c), and they shoot a bullet in front of them which is going at 10% of c. Classical physics and intuition tells us that the bullet would surely be travelling faster than light, and it’s extremely hard to get your mind around anything else that might happen. Alas, the bullet defies intuition, and not only does it not go the speed of light, but also appears to “squish” in the process (Harris 2008; Jefimenko 1998).
Special relativity, quoted from Randy Harris, is talked about sparingly, with its effects being allocated to mere optical illusions. “The truth is more challenging,” Harris wrote. When something travels through space close to the speed of light, classical physics breaks down, and you cannot simply add up the velocities to get the result that you are expecting. The newly introduced equations that could explain this phenomenon must also possess an attribute known as invariance, meaning they must hold for smaller speeds (Thompson and Cummer 2012). This set of equations can be simplified into a couple projections in two dimensions, with time (t) and x as independent variables:
Where:
Equations 1 and 2 are taking into account that the displacement and movement through time of an object in a relativistic frame of reference is calculated using the same equations as are used in Galilean methods, which is the conventional kinematics method of adding the initial displacement, and velocity multiplied by time, or the initial time and velocity multiplied by displacement, divided by the speed of light, which would give us another measure of time (Thompson 2003). Except now that convention is paired with the gamma factor (γ), which is a scalar constant that takes into account the relativistic speeds at which the system is being viewed (Harris 2008; Equation 3).
Equations 1-3 can be used to derive the formula for telling what speed objects are moving according to a relativistic frame of reference by dividing the first formula by the second, and using the fact that V′ = dx′/dt′ and V = dx/dt:
Where v is the velocity of a body travelling in reference to the inertial perspective, and V’ is the velocity of an object the body throws from their perspective, which in our example would be akin to watching a person shoot a bullet while travelling near the speed of light (Thompson 2003; Thompson and Cummer 2012; Harris 2008).
Now that we have an equation set up, we are able to input numbers and get an output which would prove the hypothesis. We know that v=0.99c, which is akin to saying “99% the speed of light”, V’=0.1c, since it is the speed of the bullet they are shooting, and c=299 792 458 m/s (CERN 2025). We can input these numbers into equation 4 and see if the bullet is actually travelling at c:
It is apparent that the bullet in reference to us would be travelling extremely close to c, and yet it would not be just quite there. It is important to note that when and V’ are small, in reference to velocities that any object on Earth would be travelling with, the value of c remains constant, and since it is much larger than the product of the two velocities, the denominator of the entire equation is essentially stated as 1, which supports the Galilean methods of conventional kinematics of adding the two velocities together (Thompson 2003; Thompson and Cummer 2012).
When working on these equations, Hendrik Lorentz was wanting to increase the invariance in his final product so as to prove the repeatability and validity of his equations, but in the end came up with the same derivation and answer as Albert Einstein did when he derived the formulae from the Maxwell-Heaviside equations in another paper (Onoochin 2022; Lorentz 1937; Poincaré 1906). Albert Einstein was the first to theorize and talk about the relativistic length contraction, which is now denoted as the “Lorentz contraction” (Jefimenko 1998). It essentially states that an object travelling at such relativistically high speeds would contract in length when being viewed from a separate frame of reference, and would itself witness other objects contract in length when viewed from its own perspective; not just in the visual sense, but in reality (Smarandache 2013; Jefimenko 1998; Harris 2008; Figure 1).
While not frequently taught in depth, and only ever used in passing for fun facts, strange phenomena occurring at high speeds is an interesting branch of physics that deserves to be known in its own glory. The expansion of scientific endeavours and feats achieved by scientists such as this has opened doors for new ideas, and a concept cool as the Lorentz transformations in all their depth of time dilation, length contraction, and many more, should not be overlooked purely because of its complexity, but rather enthused upon because of its ability to stretch the mind out of its comfort zone of intuition.
References
CERN. 2025. “Speed of Light | CERN.” https://home.cern/tags/speed-light.
Harris, Randy. 2008. “Modern Physics Second Edition (Pearson International Edition) – Anna’s Archive.” Anna’s Archive. https://annas-archive.org/md5/7bdf6228465aa2352970494e3ba23b06.
Jefimenko, Oleg D. 1998. “On the Experimental Proofs of Relativistic Length Contraction and Time Dilation.” Zeitschrift Für Naturforschung A 53 (12): 977–82. https://doi.org/10.1515/zna-1998-1208.
Lorentz, H. A. 1937. “Electromagnetic Phenomena in a System Moving with Any Velocity Smaller than That of Light.” (Dordrecht), 172–97. https://doi.org/10.1007/978-94-015-3445-1_5.
Onoochin, Vladimir. 2022. “(PDF) LORENTZ’S WORK OF 1904 AND THE LORENTZ TRANSFORMATIONS.” ResearchGate, July. https://www.researchgate.net/publication/362226387_LORENTZ S_WORK_OF_1904_AND_THE_LORENTZ_TRANSFORMATIONS.
Poincaré, M. H. 1906. “Sur la dynamique de l’électron.” Rendiconti del Circolo Matematico di Palermo (1884-1940) 21 (1): 129–75. https://doi.org/10.1007/BF03013466.
Smarandache, Florentin. 2013. “Oblique-Length Contraction Factor in Special Relativity.” 2013 (May): D1.002. https://ui.adsabs.harvard.edu/abs/2013APS..DMP.D1002S.
Thompson, Kathleen A. 2003. “Relativity, Special.” In Encyclopedia of Physical Science and Technology (Third Edition), edited by Robert A. Meyers. Academic Press. https://doi.org/10.1016/B0-12-227410-5/00657-8. Thompson, Robert T., and Steven A. Cummer. 2012. “Chapter 5 – Transformation Optics.” In Advances in Imaging and Electron Physics, edited by Peter W. Hawkes, vol. 171. Advances in Imaging and Electron Physics. Elsevier. https://doi.org/10.1016/B978-0-12-394297-5.00005-2.
Comments
7 Responses to “Shoot a Bullet at the Speed of Light! Or Not. – Special Relativity and Its Effects”
Hello Vihaan,
Great post overall and a fascinating choice of topic! Just a few recommendations:
– ““The truth is more challenging,” is what Randy Harris wrote in his 2nd edition textbook for modern physics.” is accurate, though I would rephrase slightly and add the date. Something like “As Randy Harris notes in his modern physics (2n ed., date), “The truth is more challenging””. I am not sure if writing editions changes formats based on citation style, so I would double check how to format properly.
– Make sure to add that comment about where you got this idea and how it relates to multiple disciplines!
– First sentence paragraph one would read smoother as “Imagine watching a person travel…”
Overall great read! Best of luck with editing!
Samuel Vanhaverbeke
Hey Sam!
Thank you for the suggestions!
– I don’t agree with your first suggestion fully. If you have stated the name and date of a citation previously, you don’t need to add anything but their name if mentioned with a quote within the text afterwards. If I was taking information from the book other than the quote, I would have stated more than that information. That being said, your comment was helpful in reminding me that I do not need to add anything other than his name if mentioning him after in-text citing prior. I have also made it easier to tell what quotes and information I am extracting from his work.
– I previously got lazy with the caption so I have added it now, thank you for reminding!
– I have made the change.
Thank you so much!
Hey iSci!
After learning about complex matrix multiplications and researching about their implications in the real world, along with trying to get a deeper understanding of the kinematics of relativity I learned in first year physics, I stumbled upon special relativity, and found it immensely interesting. This post is influenced by one of the most interesting phenomenon discussed in special relativity, and because of its nature defying logic as one’s intuition is trained in, I really wanted to write something about it. I hope you enjoy!
– Vihaan
Hi Vihaan,
This was a really interesting blog post. I have left a few comments below which I hope you find helpful in the editing stage:
– The current version appears to be about 800 words in length. Consider trying to get in the range of 400-500 words. I think that this can be achieved by adding equation captions. For reference, consider moving portions of paragraphs 3 and 5 into equation captions.
– If possible, consider adding a relevant image, plot, or schematic, beyond equations to grab viewers attention.
– Consider replacing “Works Cited” with” References” as per Chicago Manual of Style 18th edition.
Overall, great job! I look forward to reading the final version.
Lucas M
Hey Lucas!
Thank you so much for your suggestions.
I tried to get the post to cut in length, and it is already a lot shorter than I have in the past, but I don’t think it can get much shorter than this. I will look over and try to cut out some fillers though. Also, I don’t think equations can have figure captions.
I have added an image to display the length contraction visually. This was an excellent suggestion, thank you! I have also made the change to “works cited”
Thank you for reading!!
Cheers!
Hi Vihaan, nicely done with your blog post, your topic is very interesting! Below are a few suggestions to help improve your blog:
P1S1: I would move (c) to be right after ‘the speed of light’.
P2S2: Instead of ‘he’ I would write ‘Harris’, for clarity.
P2S3: I would reword this sentence, for instance: “The newly introduced equations that could explain this phenomenon also have to be invariant, meaning they must hold for smaller speeds (Thompson and Cummer 2012).”.
P3S1: When referencing an equation labeled as (1), the preferred method is to use “Equation 1” or simply “(1)”.
P3S1: I suggest writing out ‘multiplied by’ instead of ‘*’.
P9S1: I would say “While not frequently taught” instead of “While not taught in a very high amount”.
Overall very well done!
Happy editing,
Aven 🙂
Hey Aven!
Thank you for your suggestions!
c is the variable used for the speed of light in a vacuum, since the speed of light can vary according to what medium it is in, but thank you for suggesting!
I have made the change for every other suggestion you had.
Thank you so much for reading!
Cheers!