your name
You can work in groups of 4 for this quiz. You should turn in one quiz for your group, with all four names. You may consult the internet, notes, or books to your heart’s content.
Consider a particle of mass m in an attractive potential,
V ( r ) = −
α r
.
We had derived the following form for a trajectory, r =
1 mα/L 2 + A cos( θ − θ
0
)
.
For all the following problems consider the case where θ
0
θ = 0 .
= 0 , i.e. the minimum radius occurs at
1. (2pts) Considering the case where A > mα/L
2 energy in terms of A, m, L and α .
(the scattering problem), express the total
2. (1 pt) If one rewrites the trajectory as r = r
0
1 + β cos θ
, express r
0 and β in terms of A, m, L and α .
3. (2pts) Express the trajectory as an equation in terms of x and y instead of r and θ . Show that the expression is a hyperbola, i.e., the form is
( x − x
0
)
2 a 2
− y
2 b 2
= 1 .
Express x
0
, a and b in terms of r
0 and β .
Solutions :
1)
2) r min
=
1 mα/L 2 + A
,
E = −
α r min
+
L
2
2 mr
2 min
,
= − α ( mα/L
2
+ A ) +
L 2
2 m
( mα/L
2
+ A )
2
= − mα
2
2 L 2
+
A
2
L
2
.
2 m r r =
0
= r
0
1 + β cos θ
,
L
2 mα
, β =
AL
2 mα
= Ar
0
.

3)
1 = r
0 r + βx
,
(1 − x
2
(1 − β
2
β
2
) x +
) + 2 βr
1
βr
−
0
β 2 x
2
0 r = r
0
− βx
+ y
2 x + y
2
+ y
2
2
= r
2
0
= r
2
0
,
= r
2
0
−
+
2 βxr
0
+
β
2 r
2
0
(1 − β 2 )
,
β
2 x
2
,
(1 − β
2
) x +
βr
0
1 − β 2
2
+ y
2 x
0
= r
2
0
1 − β 2
,
= r
0
β
β 2 − 1
,
( x − x
0
)
2 a 2
− y
2 b 2
= 1 , a b
2
2
=
= r
2
0
(1 − β 2 ) 2
, r
2
0
β 2 − 1
.
Note that for a hyperbola β > 1 , whereas for an ellipse, β < 1 .