📚 Investigation 3: Instantaneous Speed | 探究3：瞬时速度
Understanding motion is one of the central themes of calculus. In this investigation, we explore how the idea of an average speed over a time interval naturally leads to the concept of instantaneous speed at a specific moment — the core idea behind derivatives.
理解运动是微积分的中心主题之一。在本次探究中，我们将探索时间间隔内的平均速度如何自然地引出某一特定时刻的瞬时速度概念——这是导数背后的核心思想。
1. Average Speed and Instantaneous Speed | 平均速度与瞬时速度
When an object moves, its average speed over a time interval [t₁, t₂] is the total distance travelled divided by the time taken: vav = Δs/Δt = (s(t₂) − s(t₁)) / (t₂ − t₁).
当物体运动时，在时间区间[t₁, t₂]内的平均速度等于总移动距离除以所用时间：vav = Δs/Δt = (s(t₂) − s(t₁)) / (t₂ − t₁)。
However, this average tells us nothing about how fast the object is moving at a single instant. For example, a car may stop and start; its average speed over one hour might be 50 km/h, but the instantaneous speed when passing a speed camera is what matters.
然而，这个平均值无法告诉我们物体在某一瞬间的运动快慢。例如，一辆汽车可能走走停停；一小时内的平均速度可能是50公里/小时，但通过测速摄像头时的瞬时速度才是关键。
2. Setting up the Investigation: Position Function | 设置探究：位置函数
We consider a particle moving along a straight line. Its position at time t is given by a function s(t), where s is measured in metres and t in seconds. A common model in IB investigations is the free-fall equation s(t) = 4.9t² (assuming no air resistance and an initial velocity of 0).
我们考虑一个沿直线运动的质点。它在时刻t的位置由函数s(t)给出，s的单位为米，t的单位为秒。IB探究中常用的一个模型是自由落体方程 s(t) = 4.9t²（假设无空气阻力且初速度为0）。
For this investigation, we aim to find the speed of the particle at exactly t = 3 seconds.
在此探究中，我们的目标是求质点在 t = 3 秒时的速度。
3. Computing Average Speeds | 计算平均速度
To approximate the instantaneous speed, we begin by computing the average speed over successively smaller time intervals starting at t = 3. For example, from t = 3 to t = 3.1, Δt = 0.1 s.
为了逼近瞬时速度，我们首先计算以 t = 3 为起点、时间间隔逐渐缩小的平均速度。例如，从 t = 3 到 t = 3.1，Δt = 0.1 秒。
vav = (s(3.1) − s(3)) / 0.1. With s(t) = 4.9t², s(3) = 4.9×9 = 44.1 m, s(3.1) = 4.9×9.61 = 47.089 m, so vav = (47.089 − 44.1)/0.1 = 29.89 m/s.
vav = (s(3.1) − s(3)) / 0.1。利用 s(t) = 4.9t²，s(3) = 4.9×9 = 44.1 m，s(3.1) = 4.9×9.61 = 47.089 m，因此 vav = (47.089 − 44.1)/0.1 = 29.89 m/s。
4. Shrinking the Time Interval | 缩短时间间隔
Next we try a smaller Δt, say 0.01 s, from t = 3 to t = 3.01. s(3.01) = 4.9×(3.01)² = 4.9×9.0601 = 44.39449 m. Then vav = (44.39449 − 44.1)/0.01 = 29.449 m/s.
接下来我们尝试更小的 Δt，例如 0.01 秒，从 t = 3 到 t = 3.01。s(3.01) = 4.9×(3.01)² = 4.9×9.0601 = 44.39449 m。那么 vav = (44.39449 − 44.1)/0.01 = 29.449 m/s。
We can continue with Δt = 0.001 s: s(3.001) = 4.9×9.006001 = 44.1294049 m, giving vav = 29.4049 m/s.
我们可以继续用 Δt = 0.001 秒：s(3.001) = 4.9×9.006001 = 44.1294049 m，得到 vav = 29.4049 m/s。
5. Observing the Trend: Convergence to a Value | 观察趋势：向某值收敛
As Δt gets smaller and smaller, the average speed values appear to approach a certain number. Let’s tabulate the results:
随着 Δt 越来越小，平均速度值似乎逼近某个确定数值。我们将结果列表如下：
| Δt (s) | s(3+Δt) (m) | Average Speed (m/s) |
|---|---|---|
| 0.1 | 47.089 | 29.89 |
| 0.01 | 44.39449 | 29.449 |
| 0.001 | 44.1294049 | 29.4049 |
| 0.0001 | 44.10294 | 29.4005 (approx) |
The average speed appears to be converging to roughly 29.4 m/s. This limiting value is the instantaneous speed at t = 3.
平均速度似乎收敛到大约 29.4 m/s。这个极限值就是在 t = 3 时的瞬时速度。
6. The Concept of a Limit | 极限的概念
Mathematically, instantaneous speed is defined as the limit of the average speed as Δt approaches zero: v(3) = limΔt→0 (s(3+Δt) − s(3)) / Δt.
从数学上讲，瞬时速度定义为当 Δt 趋近于零时平均速度的极限：v(3) = limΔt→0 (s(3+Δt) − s(3)) / Δt。
This limit, if it exists, gives the exact rate of change of position with respect to time at that instant. The notation ‘lim’ is fundamental in calculus.
如果该极限存在，它就给出了在该时刻位置相对于时间的精确变化率。记号’lim’在微积分中是基础性的。
7. The Derivative as Instantaneous Speed | 导数作为瞬时速度
In more general terms, the derivative of the position function s(t) with respect to t is s'(t) = limΔt→0 (s(t+Δt) − s(t)) / Δt. Thus instantaneous speed is just the derivative s'(t) at the specific time.
更一般地，位置函数 s(t) 关于 t 的导数是 s'(t) = limΔt→0 (s(t+Δt) − s(t)) / Δt。因此，瞬时速度就是特定时刻的导数 s'(t)。
For our example s(t)=4.9t², we can find s'(t) using the power rule: s'(t) = 9.8t. Then at t=3, s'(3)=29.4 m/s, which matches our numerical investigation.
对于我们的例子 s(t)=4.9t²，我们可以用幂法则求导：s'(t) = 9.8t。那么在 t=3 时，s'(3)=29.4 m/s，这与我们的数值探究吻合。
This demonstrates that the numerical approach to finding instantaneous speed is a valid way of approximating a derivative.
这证明了用数值方法求瞬时速度是逼近导数的一种有效方式。
8. Geometric Interpretation: Tangent Slope | 几何解释：切线斜率
Graphically, average speed over [t, t+Δt] corresponds to the slope of the secant line through points (t, s(t)) and (t+Δt, s(t+Δt)). As Δt shrinks, the secant line approaches the tangent line at t, and its slope becomes the derivative.
从图形上看，[t, t+Δt] 上的平均速度对应于经过点 (t, s(t)) 和 (t+Δt, s(t+Δt)) 的割线的斜率。当 Δt 缩小时，割线趋于 t 处的切线，其斜率即为导数。
Thus, instantaneous speed is the slope of the tangent to the position–time graph at that instant.
因此，瞬时速度就是位移–时间图像上该点切线的斜率。
9. Instantaneous Speed and Velocity | 瞬时速率与速度
In physics, speed is a scalar quantity (magnitude), while velocity is a vector. In one-dimensional motion, the instantaneous velocity can be positive or negative, indicating direction. The instantaneous speed is the absolute value of velocity.
在物理学中，速率是标量（大小），而速度是矢量。在一维运动中，瞬时速度可正可负，表示方向。瞬时速率是速度的绝对值。
In our investigation, we have been dealing with speed in the sense of velocity magnitude because the motion is in one direction. However, IB students should distinguish between the two when interpreting sign.
在本次探究中，我们处理的速度实际上是速度的大小，因为运动是单方向的。但IB学生在解读符号时应当区分两者
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