📚 Investigation 2 - Instantaneous Speed | 调查2 - 瞬时速度
In calculus and physics, the concept of instantaneous speed is central to understanding motion. Unlike average speed, which gives a broad view over an interval, instantaneous speed captures the rate of change at a specific moment. This investigation guides you through the process of approximating instantaneous speed by shrinking time intervals, culminating in the formal definition of the derivative.
在微积分和物理学中，瞬时速度的概念对于理解运动至关重要。与平均速度不同，瞬时速度捕捉了某一特定时刻的变化率。这项调查将引导你通过缩小时间间隔来逼近瞬时速度，最终引出导数的正式定义。
1. Introduction and Objectives | 引言与目标
The aim of this investigation is to explore how instantaneous speed can be determined from experimental or simulated data of a moving object. We will use the limit of average speeds as the time interval approaches zero, providing a concrete foundation for the derivative.
这项调查的目标是探索如何通过一个运动物体的实验或模拟数据来确定瞬时速度。我们将使用时隙趋近于零时平均速度的极限，为导数提供具体的奠基。
This investigation aligns with the IB Mathematics exploration criteria, encouraging the use of technology, graphical analysis, and critical reflection on the nature of limits and rates of change.
这项调查符合IB数学探索的标准，鼓励使用技术、图形分析，并对极限和变化率的本质进行批判性反思。
2. Experimental Setup: Measuring Motion | 实验设置：测量运动
To investigate instantaneous speed, we consider a freely falling object, such as a ball dropped from rest. Using a motion sensor or video tracking software, we record the distance fallen at various instants of time. The data is collected under controlled conditions, assuming negligible air resistance for simplicity.
为了研究瞬时速度，我们考虑一个自由下落的物体，比如从静止状态释放的小球。使用运动传感器或视频追踪软件，记录不同时刻的下落距离。数据在可控条件下收集，为简单起见假设空气阻力可忽略。
f real data contains small errors, this model helps illustrate the limiting process that defines instantaneous speed.
自由落体的理论模型是 s(t) = ½gt²，其中 g = 9.8 m/s²。即使真实数据包含微小误差，该模型也能帮助阐明定义瞬时速度的极限过程。
3. Data Collection: Distance vs. Time | 数据收集：距离与时间
ustration, we use a set of measurements from a simulated experiment based on s(t) = 4.9t²:
记录的位置在表格中呈现。为了说明，我们使用一组基于 s(t) = 4.9t² 的模拟实验测量数据：
| Time t (s) | Distance s (m) |
|---|---|
| 0 | 0 |
| 0.1 | 0.049 |
| 0.2 | 0.196 |
| 0.3 | 0.441 |
| 0.4 | 0.784 |
| 0.5 | 1.225 |
| 0.6 | 1.764 |
| 0.7 | 2.401 |
| 0.8 | 3.136 |
| 0.9 | 3.969 |
| 1.0 | 4.900 |
pute average speeds over intervals that shrink around a chosen instant, such as t = 0.4 s.
该数据集将用于计算在所选时刻（如 t = 0.4 s）附近逐渐缩小区间上的平均速度。
4. Calculating Average Speeds | 计算平均速度
Average speed over a time interval [t, t+Δt] is computed as vavg = (s(t+Δt) – e, between t = 0.2 s and t = 0.3 s, the distance changes from 0.196 m to 0.441 m, giving vavg = (0.441 – 0.196) / 0.1 = 2.45 m/s.
在时间间隔 [t, t+Δt] 上的平均速度计算为 vavg = (s(t+Δt) – s(t)) / Δt。例如，在 t = 0.2 s 和 t = 0.3 s 之间，距离从 0.196 m 变为 0.441 m，得出 vavg = (0.441 – 0.196) / 0.1 = 2.45 m/s。
, using different values of Δt. The forward difference quotient values are summarized in the table below.
我们可以围绕目标时刻（如 t = 0.4 s）计算多个平均速度，使用不同的 Δt 值。向前差商的值总结在下表中。
| Δt (s) | Average speed (m/s) |
|---|---|
| 0.1 | 4.41 |
| 0.05 | 4.165 |
| 0.02
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