提要 6：如何建立數學模式？(五)

(1)

提要 6：如何建立數學模式？(五)

茲以自由落體運動說明之，請參考紅色框線內之說明。

Notice:

1. 在地球表面上，物體自由落下之運動加速度約為 9.81m sec² 。 2. ₂ 9.81

2 =

dt y

d 為自由落體運動定律的化身。

3. 本問題之數學模式為二階(2 次微分)之常微分方程式，故其解會出現兩個積分常數。

4. 因有兩個積分常數會出現在通解中，故需安排兩個初始條件解之。

Physical System (實際系統)

試推算自由落體運動中，物體之移動距離與移動時間的關係。

Modeling (模式化)

由自由落體試驗得知，在地球表面附近，物體之運動加速度為

sec2

81 .

9 m 。若考慮任意時刻t(秒)

時，物體之位移量為y(公尺)，則物

體之運動加速度可以位移量y

( )

t 對時間t 的兩次微分表示之，即

81 .

2 9

2 =

dt y d

在初始條件的考慮上，可定位移起始點為y ，即₀ y

( )

0 = y₀；另外可考慮物體之初始速度為v ，因物體之₀ 運動速度可表為

dt

dy，故另一初始條

件可表為

( )

0

0 v dt

dy = 。以上兩個初始

條件即可用以解出將來通解中所出現的兩個積分常數C₁、C₂。 Mathematical Model

(數學模式) 1. 控制方程式： ₂ 9.81

2 =

dt y d

2. 初始條件：y

( )

0 = y₀

( )

0

0 v dt dy =

Solution(解)

0 0 2

2

1gt v t y

y= + +

(請參考提要 8 例題 3 之說明) Solution

Methods (解題方法) 可採用直接積分法求解。

10

(2)

習題

1. Air containing 0.06% CO2 is pumped into a room whose volume is 8,000 m³. The rate at which the air is pumped in is 2,000 m³/min, and the circulated air is then pumped out at the same rate. If there is an initial CO2 concentration of 0.2%, find the differential equation for ^{A t}

( )

, the amount (in m³) of CO2 in the room at time t.【90 交大物理所 15%】

2. Find a curve in xy plane that passes through

( )

^0,3 and whose tangent line at a point

( )

^{x y}^, has slope 2x y² .【88 成大製造所 10%】

3. Suppose that a mothball loses volume by evaporation at a rate proportional to its instantaneous area. If the diameter of the ball decreases from 2 cm to 1 cm in 3 months, how long will it take until the ball has practically gone, say, until its diameter is 1 mm.

【87 中原化工所 20%】

4. An oil tanker of mass M is sailing in a straight line. At time zero it shut off its engines and coasts. Assume that the water tends to slow the tanker with a force proportional to

( )

^m

v t 

  , in which ^{v t}

( )

is the velocity at time t and m is constant.

(a) Derive a differential equation for ^{v t}

( )

and solve it. (Let v v= at time zero.) ₀ (b) Does the tanker eventually come to a full stop? If so, find the time required for the

tanker to stop. If not, why?

Hint: Solve and discuss the problem by considering cases on m. (i) 0< <m 1 (ii) 1

m= (iii) m>1.【89 台大化工所 15%】

11