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《动力学中的临界与极值问题》进阶练习(一)

《动力学中的临界与极值问题》进阶练习(一)
《动力学中的临界与极值问题》进阶练习(一)

《动力学中的临界与极值问题》进阶练习

1.速度大小是5m/s的甲、乙两列火车,在同一直线上相向而行。当它们相隔2000m时,一只鸟以10m/s的速度离开甲车头向乙车头飞去,当到达乙车车头时立即返回,并这样连续在两车间来回飞着。问:

(1)当两车头相遇时,这鸟共飞行多少时间?

(2)相遇前这鸟飞行了多少路程?

2.在平直公路上一汽车的速度为15m/s,从某时刻汽车开始刹车,在阻力作用下,汽车以2m/s2的加速度做匀减速运动,则刹车后第10s末车离刹车点的距离是 m.

3.A、B两车停在同一点,某时刻A车以2m/s2的加速度匀加速开出,2s后B车同向以3m/s2的加速度开出。问:B车追上A车之前,在启动后多长时间两车相距最远,距离是多少?

参考答案

1.【解析】飞鸟飞行的时间即为两车相遇前运动的时间,由于飞鸟在飞行过程中速率没有变化,可用s=vt 求路程。

(1)设甲、乙相遇时间为t ,则飞鸟的飞行时间也为t ,甲、乙速度大小相等v 甲= v 乙=5m/s ,同相遇的临界条件可得:s = (v 甲+v 乙)t 则:2000=20010

s t s s v v ==+乙甲 (3) 这段时间,鸟飞行的路程为:10200s vt m '==?

2.【解析】 设汽车从刹车到停下来所用时间为t 0, 由运动学规律得:0000150,7.52

t t v v v v at t s s a --=-=== 由于t 0<10s ,所以在计算时应将t=7.5s 代入公式求解。 则有:22011(157.527.5)56.2522

s v t at m m =-=?-??= 3.【解析】〖解法1〗由于当A 车的加速度度小于B 车的加速度,B 车后启动,则B 车一定能追上A 车,在追上前当两车的速度相等时,两车相距最远。设当A 车运动t 时间时,两车速度相等,则有,(3)A B A B v v a t a t ==- 解得:39B A B

a t s a a ==- 把t 代入两车之间距离差公式得:2211(3)2722

A B A B s s s a t a t m ?=-=--= 〖解法2〗设A 启动ts 两车相距最远,A 车的位移:212

A s at =,

B 车的位移:21(3)2

B s a t =- 两车间距离为22211(3)0.5913.522

A B A B s s s a t a t t t ?=-=--=-+- 由数学知识可知,当992(0.5)

t s s =-=?-时, 两车间有最大距离:2211(3)2722A B A B s s s a t a t m ?=-=

--=

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