【CH09】Center of Mass and Momentum Homework of ... - 物理學系

Fundamentals of Physics
Halliday & Resnic
東海大學物理系
(b) The problem states that the frictional forces acting on the barges does not depend on
mass, so the loss of mass from the slower barge does not affect its motion (so no extra
force is required as a result of the shoveling).
【補充資料】關於一維完全彈性碰撞
【University Physics, Harris Benson】
【Problem 9-7】
A particle of mass 1 m moving at velocity 1 1 u = u i makes a one-dimensional elastic collision with
a particle of mass 2 m moving at velocity 2 2 u = u i. The final velocities are 1 1 v = v i and 2 2 v = v i.
( m1− m2) u1+ 2m2u2
Show that v1
=
m1+ m2
2 mu 1 1+ ( m2−m1) u2
v2
=
m + m
1 2
一個粒子(質量 m 1 ,速度 1 = u1
碰撞後兩者分別為 1 = v1
=
u i),和另一個粒子(質量 m 2 , u2 = u2i)做完全彈性碰撞,
v i和 v2 v2i.,請推導出
( m1− m2) u1+ 2m2u2
2 mu 1 1+ ( m2−m1) u2
v1
=
, v2
=
。
m1+ m2
m1+ m2
1 2 1 2 1 2 1 2
:能量守恆: mu 1 1+ mu 2 2= mv 1 1+ mv 2 2……………....○1
2 2 2 2
⇒
m ( u − v ) = m ( v − u )
2 2 2 2
1 1 1 2 2 2
m ( u − v )( u + v ) = m ( v − u )( v + u ) …...○3
⇒ 1 1 1 1 1 2 2 2 2 2
動量守恆: mu 1 1+ mu 2 2= mv 1 1+ mv 2 2…………..………….....○2
m ( u − v ) = m ( v −u ) …..............................○4
⇒ 1 1 1 2 2 2
由○3 、○4 式: u1+ v1 = u2 + v2
代回○2 式 mu 1 1+ mu 2 2= mv 1 1+ m2( u1+ v1− u2)
( m1− m2) u1+ 2 m2u2 = ( m1+ m2) v1
( m1− m2) u1+ 2m2u2
v1
=
m + m
1 2
Fundamentals of Physics
Halliday & Resnic
2 mu + ( m − m) u
1 1 2 1 2
同理 v2
=
m1+ m2
【補充資料】關於二維完全彈性碰撞
( m − m ) u + 2m
u
v1
=
m1+ m2
2 mu + ( m − m) u
v2
=
m + m
1 2 1 2 2
1 1 2 1 2
1 2
東海大學物理系