问题 问答题

如图所示,光滑水平面上,质量为2m的小球B连接着轻质弹簧、处于静止状态;质量为m的小球A以速度v0向右匀速运动,接着逐渐压缩弹簧并使B运动,过一段时间后,A与弹簧分离.设小球A、B与弹簧相互作用过程中无机械能损失,弹簧始终处于弹性限度以内.

(1)求当弹簧被压缩到最短时,弹簧的弹性势能E.

(2)若开始时在小球B的右侧某位置固定一块挡板(图中未画出),在小球A与弹簧分离前使小球B与挡板发生正撞,并在碰后立刻将挡板撤走.设小球B与固定挡板的碰撞时间极短,碰后小球B的速度大小不变、但方向相反.设此后弹簧性势能的最大值为Em,求Em可能值的范围.

答案

(1)当A球与弹簧接触以后,在弹力作用下减速运动,而B球在弹力作用下加速运动,弹簧势能增加,当A、B速度相同时,弹簧的势能最大.

设A、B的共同速度为v,弹簧的最大势能为E,则A、B系统动量守恒:mv0=(m+2m)v①

由机械能守恒:

1
2
m
v20
=
1
2
(m+2m)v2+E…②

联立两式得:E=

1
3
m
v20
…③

(2)设B球与挡板碰撞前瞬间的速度为vB,此时A的速度为vA

系统动量守恒:mv0=mvA+2mvB…④

B与挡板碰后,以vB向左运动,压缩弹簧,当A、B速度相同(设为v)时,弹簧势能最大,为Em

则:mvA-2mvB=3mv…⑤

1
2
m
v20
1
2
×3m
v2共
+Em…⑥

由④⑤两式得:v=

v0-4vB
3
代入⑥式,化简得:Em=
8m
3
[-(vB-
v0
4
2+
3v20
16
]…⑦

而当弹簧恢复原长时相碰,vB有最大值vBm,则:

mv0=mvA′+2mvBm          

1
2
mv02=
1
2
mvA2+
1
2
×2mvBm2

联立以上两式得:vBm=

2
3
v0即vB的取值范围为:0<VB
2
3
v0…⑧

结合⑦式可得:当vB=

v0
4
时,Em有最大值为:
1
2
m
v20
…⑨

当vB=

2
3
v0时,Em有最小值为:
1
27
m
v20

答:(1)当弹簧被压缩到最短时,弹簧的弹性势能是

1
3
m
v20

(2)此后弹簧的弹性势能最大值的范围为[

1
27
m
v20
1
2
m
v20
].

阅读理解

Olympic torch relay (奥运火炬接力) planned route

Beijing, April 27—The Beijing 2008 Olympic Torch Relay will cover the greatest area and include the largest number of people. The plans were announced on Thursday night. The planned route would last  130 days and travel 137 000 km.

First Nobel Prize winner to donate to Hope School

Chinanews, Beijing, May 10—Professor Dannel Chee Tsui, in the USA, signed the agreement to donate 350 000 yuan to his home town to build a Hope Primary School in China.

Bill Gates receives Tsinghua honorary doctorate

Beijing,April 20—Bill Gates, chairman of global IT giant Mi­crosoft, received an honorary doc­torate (荣誉博士学位) when he

visited China’s   famous   Tsinghua   University   on Thursday.

Big Shaolin kung fu center to be built in Russia

Chinanews, Beijing, April 27—China and Russia have signed a letter of intent ( 意向书)on the building of a Shaolin kung fu center in Russia. It will be the first of its kind in Russia,and also the biggest overseas kung fu center when it is completed.

小题1: How long will the Beijing 2008 Olympic Torch Relay last?

A.One month.

B.Two months.

C.Three months.

D.More than four months.小题2: Who signed an agreement to donate money to build a Hope Primary School on May 10?

A.Bill Gates.

B.Professor Daniel Chee Tsui.

C.Beijing government.

D.Tsinghua University.小题3: What will be built in Russia according to the news?

A.A university.

B.A sports centre.

C.A Shaolin kung fu center.

D.A Hope Primary School.

选择题