Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

15.20: Acceleration

( \newcommand{\kernel}{\mathrm{null}\,}\)

Figure XV.33 shows two references frames, Σ and Σ, the latter moving at speed v with respect to the former.

alt

A particle is moving with acceleration a in Σ. (“ in Σ ” = “referred to the reference frame Σ ”.) The velocity is not necessarily, of course, in the same direction as the acceleration, and we’ll suppose that its velocity in Σ is u. The acceleration and velocity components in Σ are ax,ay,ux,uy.

What is the acceleration of the particle in Σ? We shall start with the x-component.

The x-component of its acceleration in Σ is given by

ax=duxdt,

where

ux=ux+v1+uxvc2

and

t=γ(t+vxc2)

Equations 15.16.2 and 15.5.19 give us

dux=duxduxdux=duxγ2(1+uxvc2)2

and

dt =ttdt+txdx=γdt + γvc2dx

On substitution of these into Equation ??? and a very little algebra, we obtain

ax=aγ3(1+uxvc2)3

The y-component of its acceleration in Σ is given by

ay=duydt,

We have already worked out the denominator dt(Equation ???). We know that

uy=uyγ(1+uxvc2)

from which

duy=uyux+uyuyuy=1γ(vuyc2(1+vuxc2)2dux+11+vuxc2duy).

Divide Equation ??? by Equation ??? to obtain

ay=1γ2(vuyc2(1+vuxc2)2ax+11+vuxc2ay).


This page titled 15.20: Acceleration is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?