Relativistic Force

Let there be a given mass mm, acted by a net force FF. If the mass travels a small distance drdr, prove that the infinitesimal change in energy dEdE equals the work done by FF.

Note that m,Em, E are scalars while F,drF, dr are vectors.

Solution

We begin by defining F=dpdtF = \frac{dp}{dt} and dr=vdtdr = v dt, where vv is the velocity-vector.

Thus, Fdr=dpdtvdt.Fdr = \frac{dp}{dt}\cdot v dt.

In relativistic mechanics, E=(pc)2+(mc2)2E = \sqrt{{(pc)}^{2} + {(m{c}^{2})}^{2}}

hence,

dEdt=pc2(pc)2+(mc2)2dpdt\frac{dE}{dt} = \frac{p{c}^{2}}{\sqrt{{(pc)}^{2} + {(m{c}^{2})}^{2}}} \cdot \frac{dp}{dt}

This long expression can be reduced to pc2Edpdt\frac{p{c}^{2}}{E} \cdot \frac{dp}{dt}, which can be further reduced to vdpdtv\cdot \frac{dp}{dt}.

Assembling the above results yield dEdt=vdpdt\frac{dE}{dt} = v\cdot\frac{dp}{dt}.

Therefore, dE=Fdr.dE = F\cdot dr.

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
5 years, 3 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Elegant!! Good job

Racchit Jain - 3 years, 9 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...