Classical Mechanics LECTURE 7: TWO-BODY ELASTIC COLLISIONS
Classical Mechanics
LECTURE 7:
TWO-BODY ELASTIC COLLISIONS
Prof. N. Harnew University of Oxford
MT 2016
1
OUTLINE : 7. TWO-BODY ELASTIC COLLISIONS
7.1 Two-body collisions - general concepts
7.1.1 Momentum exchange and impulse 7.1.2 An off-axis collision in 2D
7.2 Elastic collisions in the Lab frame
7.2.1 Elastic collisions in 1D in the Lab frame 7.2.2 Special case in 1D where target particle is at rest 7.2.3 Collision in 2D : equal masses, target at rest
2
7.1 Two-body collisions - general concepts
No external forces. Collision via massless springs or other force type.
ti : collision starts. All
energy is kinetic.
Ti
=
1 2
m1
u12
+
1 2
m2u22
t : collision peaks. Some
kinetic is converted into
potential (of the spring).
E
=
1 2
m1x12
+
1 2
m2x22
+
Eint
tf : collision ends. All
energy is kinetic again.
Tf
=
1 2
m1v12
+
1 2
m2v22;
Ti = Tf + E ( inelastic)
3
7.1.1 Momentum exchange and impulse
During collision: internal force causes change of momentum
F
=
dp dt
At ti : total momentum
p = p1 + p2 = m1u1 + m2u2
At t : m1 d p1 = F12dt
m2 d p2 = F21dt
At tf : m1 v1 and m2 v2
Impulse p1 = I1 =
tf ti
F12dt
p2 = I2 =
tf ti
F21dt
Since F12 + F21 = 0 p1 + p2 = 0
Momentum conserved.
4
7.1.2 An off-axis collision in 2D
Impulse is along line of
centres
p1 =
tf ti
F12dt
p2 =
tf ti
F21dt
v1
=
1 m1
tf ti
F12dt
+
u1
5
v2
=
-
1 m2
tf ti
F12dt
................
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