數學differentiation

Taiman Chan

2011-04-02 7:15 pm

第一條：

Suppose the brakes of a car produce a constant deceleration (i.e. an acceleration in the opposite direction of velocity) of a m/s square

(a) If the car travelling at 16m/s will come to stop in 20m after the brakes are applied, find the value of a.

(b) With the same value of a in (a), how far will the car move before being brought to stop if it is travelling at 32m/s?

（請用數學differentiation,勿用物理學公式）
第二條：

圖片參考：http://i784.photobucket.com/albums/yy122/linguisticsprofessor/0aabcfdb.jpg?t=1301713670

回答 (1)

myisland8132

2011-04-02 8:50 pm

✔ 最佳答案

1(a) dv/dt = a

v = v_0 + at

s = v_0t + (1/2)at^2

Sub. s = 20, v_0 = 16 and v = 0

20 = 16t - 8t => t = 2.5s

a = -1/2.5 = -6.4 m/s^2

(b) 0 = 32 - 6.4t => t = 5

s = 32*5 + (1/2)(-6.4)(5^2) = 160 - 80 = 80m

2(a) Let the radius of the outer circle is R and the radius of the inner circle is r. Then dR/dt = 2 and dr/dt = 3.

Since A = π(R^2 - r^2). dA/dt = 2π(R*dR /dt - r*dr/dt) = 2π(2R - 3r)

If 2R - 3r < 0 or r >(2/3)R, A will begin to decrease.

We just obtain the equation 10 + 3t > (2/3)[40 + 2t] or 30 + 9t > 80 + 4t => t > 10

After 10s, A will begin to decrease.

(b) dA/dt = 0 => r = (2/3)R and t = 10. At this time r = 40, R = 60 and A = π(R^2 - r^2) = π(60^2 - 40^2) = 2000π cm^2

(c) When r = R, the two circles will overlap. Let 10 + 3t = 40 + 2t => t = 30s

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