An Exploration on the Effect of the Launch Angle of a Projectile on the Horizontal Range | 抛体发射角对水平射程影响的探究

📚 An Exploration on the Effect of the Launch Angle of a Projectile on the Horizontal Range | 抛体发射角对水平射程影响的探究

Projectile motion is a cornerstone of classical mechanics, and the effect of launch angle on horizontal range is a classic topic for investigation. In the IB Physics Internal Assessment (IA), students are often required to design and conduct an experiment that explores this relationship while carefully controlling variables and analysing uncertainties. This article provides a comprehensive walkthrough of such an exploration, covering the theoretical foundation, experimental design, data processing, and evaluation, aligned with the IB criteria for Personal Engagement, Exploration, Analysis, and Evaluation.

抛体运动是经典力学的基石，而发射角对水平射程的影响是一个经典的探究课题。在 IB 物理内部评估（IA）中，学生通常需要设计并实施一项实验，在仔细控制变量的同时探究这一关系，并分析不确定度。本文全面解析了这一探究过程，涵盖理论基础、实验设计、数据处理和评价，与 IB 对个人参与、探究、分析和评价的评分标准保持一致。

1. Theoretical Background | 理论基础

In an idealised model where air resistance is neglected, the only force acting on a projectile after launch is the constant downward acceleration due to gravity, g. The motion can be decomposed into independent horizontal and vertical components. The horizontal velocity remains constant at vₓ = v₀ cos θ, while the vertical velocity changes according to vᵧ = v₀ sin θ – g t.

在忽略空气阻力的理想模型中，抛体发射后唯一的作用力是恒定的向下重力加速度 g。运动可分解为独立的水平和竖直分量。水平速度 vₓ = v₀ cos θ 保持不变，而竖直速度按 vᵧ = v₀ sin θ – g t 变化。

By setting the vertical displacement to zero (launch and landing at the same height), the total time of flight is t = (2 v₀ sin θ) / g. Multiplying this by the constant horizontal velocity gives the horizontal range R. The resulting well-known equation is:

令竖直位移为零（发射与落地同高度），可得总飞行时间 t = (2 v₀ sin θ) / g。乘上恒定的水平速度即得水平射程 R。由此得出众所周知的方程：

R = (v₀² sin 2θ) / g

This equation reveals that for a fixed initial speed, the range is directly proportional to sin 2θ. The sine function reaches its maximum value of 1 when 2θ = 90°, i.e., θ = 45°. Therefore, the theoretical optimal launch angle for maximum range is 45°, and the range should be symmetric for complementary angles, such as 30° and 60°.

该方程表明，当初速度固定时，射程与 sin 2θ 成正比。当 2θ = 90°，即 θ = 45° 时，正弦函数达到最大值 1。因此，理论上获得最大射程的最佳发射角为 45°，并且对于互补角（如 30° 和 60°），射程应具有对称性。

2. Research Question and Hypothesis | 研究问题与假设

Research Question: How does the launch angle (θ) of a projectile, launched from and landing at the same vertical level, affect its horizontal range (R)?

研究问题：对于在同一水平高度发射和落地的抛体，其发射角（θ）如何影响水平射程（R）？

Hypothesis: Based on the theoretical model R = (v₀² sin 2θ) / g, the range will be proportional to sin 2θ. Therefore, as θ increases from 0° to 90°, R will increase, reach a maximum at approximately 45°, and then decrease. The relationship will be symmetric about 45°, and a plot of R versus sin 2θ should yield a straight line passing through the origin, with a gradient equal to v₀²/g.

假设：基于理论模型 R = (v₀² sin 2θ) / g，射程将与 sin 2θ 成正比。因此，随着 θ 从 0° 增大到 90°，R 将先增大，在约 45° 处达到最大值，然后减小。该关系将关于 45° 对称，并且 R 对 sin 2θ 的图像应为一条过原点的直线，其斜率等于 v₀²/g。

3. Variables and Control | 变量与控制

In this investigation, careful identification and control of variables are essential to ensure that any observed changes in the dependent variable are solely due to the manipulation of the independent variable. The following table outlines the key variables and the methods used to control them.

在本探究中，仔细识别和控制变量至关重要，以确保因变量的任何观测变化仅由自变量的操作引起。下表列出了关键变量及其控制方法。

Variable Type | 变量类型	Variable | 变量	Details / How Controlled | 详情/控制方式
Independent | 自变量	Launch angle (θ) | 发射角（θ）	Varied from 10° to 80° in 10° increments using a protractor attached to the launcher. | 借助发射器上的量角器，以 10° 为步长，从 10° 改变至 80°。
Dependent | 因变量	Horizontal range (R) | 水平射程（R）	Measured with a metre rule from the launch point to the point of first impact on the same horizontal plane. | 使用米尺测量从发射点到同一水平面上首次落点之间的水平距离。
Controlled | 控制变量	Initial launch speed (v₀) | 初始发射速度（v₀）	Maintained by using the same compression spring setting or consistent elastic band extension for each trial. The launcher was tested for consistency. | 每次试验均使用相同的弹簧压缩设置或一致的弹弓带延伸量，并对发射器的稳定性进行了测试。
Controlled | 控制变量	Mass and shape of projectile | 抛体质量与形状	Identical spherical steel ball used throughout. | 全程使用同一球形钢球。
Controlled | 控制变量	Environmental factors | 环境因素	Experiment conducted indoors with negligible air currents; temperature held approximately constant. | 实验在室内进行，气流可忽略不计；温度基本保持恒定。

4. Apparatus and Setup | 仪器与装置

The experiment requires straightforward equipment commonly available in a school physics laboratory. The assembly must ensure reproducible launches and accurate angle and distance measurements.

实验所需的设备简单，一般在中学物理实验室均可获得。装置必须确保可重复发射，并能精确测量角度和距离。

Equipment list: Projectile launcher (spring-loaded or elastic) with an angle scale, uniform steel ball (diameter ~1.5 cm), metre rule (precision ±0.1 cm), carbon paper and plain paper, clamp stand, plumb line, and level. A digital video camera (optional) can supplement measurements.

设备清单：带有角度刻度的抛体发射器（弹簧或弹弓式）、均质钢球（直径约 1.5 cm）、米尺（精度 ±0.1 cm）、复写纸和白纸、铁架台、铅垂线和水准仪。（可选）数码摄像机可辅助测量。

The launcher is firmly clamped to the table edge with the barrel aligned so that the exit is at a known height above the floor, but for the simplified experiment described here, the projectile is launched from a platform so that it lands at the same vertical level. The plumb line ensures the 0° reference is truly horizontal.

发射器被牢固地夹在桌边，枪管方向确保出口位于地板上方已知高度，但本文所述的简化实验中，抛体从平台上发射，落地时位于同一竖直高度。铅垂线确保 0° 基准真正水平。

5. Method and Procedure | 方法与步骤

English steps:

1. Set the launcher to an angle of 10° using the integrated protractor and confirm with a digital inclinometer if available. Ensure the projectile exits at the same height as the landing area.
2. Load the projectile and compress the spring to the consistent pre-marked position. Release the projectile.
3. Place a sheet of plain paper on the landing area and cover it with carbon paper (ink side down). After the impact, measure from the launch point to the mark left on the plain paper using a metre rule. Record the horizontal distance.
4. Repeat the launch three times at the same angle to obtain a set of range values and calculate the mean.
5. Repeat steps 1–4 for angles 20°, 30°, 40°, 45°, 50°, 60°, 70°, and 80°. For angles above 45°, ensure the landing area is still at the same level.
6. Throughout the measurements, keep the launcher position fixed and avoid any disturbance to the spring tension.

中文步骤：

1. 利用发射器上的量角器将其角度设定为 10°，若有数显倾角仪则进行验证。确保抛体出口与落地区域处于同一高度。
2. 装入抛体并将弹簧压缩至预设的统一标记位置。释放抛体。
3. 在落点区域放置一张白纸，上面覆盖复写纸（墨面朝下）。撞击后，使用米尺从发射点测量至白纸上的印记，记录水平距离。
4. 在相同角度下重复发射三次，获得一组射程值，并计算平均值。
5. 对 20°、30°、40°、45°、50°、60°、70° 和 80° 重复步骤 1–4。对于 45° 以上角度，确保落点区域仍在同一水平面。
6. 在整个测量过程中，保持发射器位置固定，并避免对弹簧张力造成任何干扰。

6. Data Collection | 数据收集

Raw data should be recorded in a clear table that includes the angle, three trial readings, and the mean range. An example of such a table is presented below. Students must mention the instrument uncertainties and estimation of random errors.

原始数据应记录在一个清晰的表格中，包含角度、三次试验读数和平均射程。下表为一个示例。学生需说明仪器不确定度以及随机误差的评估。

Angle θ / ° | 角度 θ / °	Trial 1 R / cm | 试验 1 R / cm	Trial 2 R / cm | 试验 2 R / cm	Trial 3 R / cm | 试验 3 R / cm	Mean R / cm | 平均 R / cm	Max. deviation / cm | 最大偏差 / cm
10	28.1	27.8	28.5	28.1	0.4
20	48.3	49.1	48.7	48.7	0.4
30	62.0	61.5	62.3	61.9	0.4
40	68.9	69.5	68.3	68.9	0.6
45	71.2	70.8	71.6	71.2	0.4
50	69.5	68.9	69.3	69.2	0.6
60	62.4	61.8	62.1	62.1	0.3
70	48.5	49.0	48.2	48.6	0.4
80	28.3	27.9	28.4	28.2	0.3

The uncertainty in the mean range is estimated from the maximum deviation from the mean. The metre rule has a reading uncertainty of ±0.05 cm, but the dominant uncertainty is the spread of the repeat trials, which accounts for variations in launch speed and landing point identification.

平均射程的不确定度通过最大偏差估算。米尺的读数不确定度为 ±0.05 cm，但主要不确定度来自重复试验的离散度，这反映了发射速度波动和落点识别的差异。

7. Data Processing and Analysis | 数据处理与分析

The primary data suggest a peak around 45° and symmetry, but a more rigorous analysis involves linearising the relationship. By plotting mean range R against sin 2θ, the model predicts a straight line through the origin with slope = v₀²/g. Values for sin 2θ are calculated for each angle:

原始数据表明峰值出现在 45° 附近且具有对称性，但更严谨的分析需对关系进行线性化。通过绘制平均射程 R 对 sin 2θ 的图像，模型预测将得到一条过原点且斜率为 v₀²/g 的直线。每个角度对应的 sin 2θ 值计算如下：

θ / °	Sin 2θ	Mean R / cm | 平均 R / cm
10	0.342	28.1
20	0.643	48.7
30	0.866	61.9
40	0.985	68.9
45	1.000	71.2
50	0.985	69.2
60	0.866	62.1
70	0.643	48.6
80	0.342	28.2

When a scatter plot of R vs sin 2θ is constructed, a clear linear trend emerges. Performing a linear regression yields a best-fit line with a high correlation coefficient (R² > 0.99) and a slope around 71.2 cm. Using g = 9.81 m s⁻², the experimental initial speed v₀ can be extracted: v₀ = √(slope × g) ≈ √(0.712 m × 9.81 m s⁻²) ≈ 2.64 m s⁻¹.

当构建 R 对 sin 2θ 的散点图时，呈现出清晰的线性趋势。进行线性回归后得到一条最佳拟合线，相关系数很高（R² > 0.99），斜率约为 71.2 cm。取 g = 9.81 m s⁻²，可反推实验初始速度 v₀：v₀ = √(斜率 × g) ≈ √(0.712 m × 9.81 m s⁻²) ≈ 2.64 m s⁻¹。

The data confirm the predictions: the points for 30° and 60° are nearly identical (61.9 cm vs 62.1 cm), as are 20° and 70° (48.7 cm vs 48.6 cm) and 10° and 80° (28.1 cm vs 28.2 cm). The slight asymmetries are within the uncertainty bounds, indicating strong agreement with the model.

数据证实了预测：30° 与 60° 的数据几乎相同（61.9 cm 对 62.1 cm），20° 与 70°（48.7 cm 对 48.6 cm）以及 10° 与 80°（28.1 cm 对 28.2 cm）也呈现出同样的对称性。微小的不对称处于不确定度范围之内，表明与模型

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