nr. 475 - 2010 - Institut for Natur, Systemer og Modeller (NSM)
nr. 475 - 2010 - Institut for Natur, Systemer og Modeller (NSM)
nr. 475 - 2010 - Institut for Natur, Systemer og Modeller (NSM)
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
6.2 Det skrå kast 95<br />
=<br />
x(t) =<br />
F<br />
m<br />
t<br />
c u0x<br />
0<br />
<br />
<br />
1 − u2 0<br />
c 2<br />
1 + ( F<br />
m c )2 1 − u2 0<br />
c2 t<br />
0<br />
F<br />
m<br />
d F<br />
m<br />
u0x dt<br />
t 2 − 2 F<br />
m c 2<br />
<br />
1 − u2 0<br />
c2 <br />
t − u0y<br />
<br />
1 − u2 0<br />
c 2 t − u0y<br />
2<br />
<br />
1 − u2 0<br />
c 2 t u0y<br />
+ c 2 − u 2 0y<br />
(6.61)<br />
Ved opslag i en integraltabel findes<br />
<br />
1<br />
√ dx = ln |x +<br />
x2 + a √ x2 + a| (6.62)<br />
Da integralet i ligning (6.61) netop er af denne <strong>for</strong>m, finder vi<br />
x(t) =<br />
F<br />
m<br />
c u0x<br />
<br />
1 − u2 0<br />
c 2<br />
ln<br />
<br />
<br />
F<br />
For y-koordinaten fås tilsvarende<br />
=<br />
−F<br />
m<br />
y(t) =<br />
c<br />
t<br />
0<br />
<br />
1 − u2 0<br />
c 2<br />
<br />
m<br />
<br />
1 − u2 0<br />
c2 2 t − u0y + c2 − u2 0y + F<br />
<br />
1 − m<br />
u2 0<br />
c2 t − u0y<br />
( −F<br />
m<br />
c − u0y<br />
<br />
1 − u2 0<br />
c 2 t + u0y) dt<br />
1 + ( F<br />
m c )2 (1 − u2 0<br />
c 2 ) t 2 − 2 F<br />
m c 2<br />
t<br />
0<br />
2<br />
<br />
d<br />
<br />
− F<br />
m<br />
− F<br />
m<br />
<br />
1 − u2 0<br />
c2 2 t + u0y<br />
<br />
1 − u2 0<br />
c 2 t + u0y<br />
2<br />
<br />
1 − u2 0<br />
c 2 t u0y<br />
+ c 2 − u 2 0y<br />
Dette integral kan umiddelbart findes, <strong>og</strong> resultatet <strong>for</strong> y(t) er<br />
y(t) =<br />
c<br />
<br />
F 1 − m<br />
u2 0<br />
c2 <br />
c −<br />
<br />
<br />
− F<br />
<br />
m<br />
1 − u20 t + u0y<br />
c2 2<br />
+ c 2 − u 2 0y<br />
Dermed er banekurven <strong>for</strong> partiklen bestemt. Se Fig. (6.3).<br />
<br />
(6.63)<br />
(6.64)<br />
(6.65)