Calculating the acceleration | 加速度的计算

📚 Calculating the acceleration | 加速度的计算

Acceleration is a central concept in A-Level Physics. It appears in kinematics, dynamics, graphs of motion and experiments. This article explains how to calculate acceleration from definitions, data, graphs and the SUVAT equations for uniform motion.

加速度是 A-Level 物理中的核心概念。它出现在运动学、动力学、运动图像和实验中。本文讲解如何根据定义、数据、图像以及匀变速运动的 SUVAT 方程来计算加速度。

1. What is Acceleration? | 什么是加速度？

Acceleration is the rate of change of velocity with respect to time. It tells us how quickly an object speeds up, slows down or changes direction. Since velocity is a vector, any change in its magnitude or direction counts as acceleration.

加速度是速度随时间的变化率。它表示物体速度大小增加、减小或方向改变的快慢。由于速度是矢量，速度的大小或方向发生任何变化都算作加速度。

For example, a car moving at a constant speed around a bend is accelerating because its direction is changing, even though its speed is constant.

例如，一辆汽车以恒定速率转弯时仍在加速，因为它的方向在改变，尽管速率不变。

2. The Defining Equation | 定义公式

The average acceleration a over a time interval Δt is calculated by dividing the change in velocity Δv by Δt. If the initial velocity is u and the final velocity is v, then:

在时间间隔 Δt 内的平均加速度 a 等于速度变化量 Δv 除以 Δt。若初速度为 u，末速度为 v，则：

a = Δv / Δt = (v − u) / t

For instantaneous acceleration, the time interval is made extremely small, so the acceleration is the derivative of velocity with respect to time: a = dv/dt.

对于瞬时加速度，时间间隔趋近于零，因此加速度是速度对时间的导数：a = dv/dt。

3. Units of Acceleration | 加速度的单位

The SI unit of acceleration is metre per second squared, written m s⁻² or m/s². This arises because velocity is measured in m s⁻¹ and time in s, so acceleration has units (m s⁻¹)/s = m s⁻².

加速度的国际单位是米每二次方秒，写作 m s⁻² 或 m/s²。因为速度的单位是 m s⁻¹，时间的单位是 s，所以加速度的单位为 (m s⁻¹)/s = m s⁻²。

In everyday situations, acceleration may also be quoted in km h⁻¹ s⁻¹, but this must be converted to SI units before using standard equations.

在日常生活中，加速度有时也用 km h⁻¹ s⁻¹ 表示，但在使用标准方程前必须转换为国际单位。

9.81 m s⁻² = 9.81 m s⁻² × (3600 s / 1000 m) = 35316 km h⁻¹ per hour? No, careful: 1 m s⁻² = 12960 km h⁻¹ per hour.

The simplest rule is to convert all speeds to m s⁻¹ and all times to seconds before substituting into equations.

最简单的规则是在代入公式之前，先将所有速度换算为 m s⁻¹，所有时间换算为秒。

4. Acceleration as a Vector | 作为矢量的加速度

Acceleration is a vector quantity, so it has both magnitude and direction. In one-dimensional motion, choose a positive direction first. A positive value of a means acceleration in that direction, while a negative value means acceleration opposite to it.

加速度是矢量，既有大小又有方向。在一维运动中，先选定正方向。正的 a 表示沿正方向加速，负的 a 表示沿正方向的反方向加速。

Negative acceleration is often called deceleration or retardation. However, in physics it is safer to say acceleration in the negative direction, because the sign depends on the chosen coordinate system.

负加速度常被称为减速或 retardation。但在物理中更严谨的说法是沿负方向的加速度，因为符号取决于所选的坐标系。

5. Calculating Acceleration from Velocity-Time Data | 从速度-时间数据计算加速度

Given velocity-time data, select two readings, calculate the change in velocity and divide by the corresponding time interval.

已知速度-时间数据时，选取两个读数，计算速度的变化量，再除以相应的时间间隔。

Time t / s	Velocity v / m s⁻¹
0	5
2	11
4	17
6	23

Using the first and last readings, the change in velocity is 23 − 5 = 18 m s⁻¹ and the time interval is 6 − 0 = 6 s.

利用第一组和最后一组数据，速度变化量为 23 − 5 = 18 m s⁻¹，时间间隔为 6 − 0 = 6 s。

a = (23 − 5) / (6 − 0) = 18 / 6 = 3.0 m s⁻²

The acceleration is constant in this example because the velocity increases by the same amount every second.

在这个例子中加速度恒定，因为速度每秒增加相同的量。

6. Using Velocity-Time Graphs | 利用速度-时间图像

On a velocity-time graph, acceleration is the gradient or slope of the graph. A straight line means constant acceleration; a curved line means changing acceleration.

在速度-时间图像中，加速度是图像的斜率。直线表示加速度恒定，曲线表示加速度在变化。

To calculate the gradient between two points on a straight line, use:

要计算直线上两点之间的斜率，可使用：

a = (v₂ − v₁) / (t₂ − t₁)

A positive gradient gives positive acceleration, a negative gradient gives negative acceleration, and a horizontal line gives zero acceleration.

斜率为正对应正加速度，斜率为负对应负加速度，水平线表示加速度为零。

For a curved velocity-time graph, the acceleration at a point is the gradient of the tangent to the curve at that point.

对于速度-时间曲线，某一点的加速度等于曲线在该点切线的斜率。

7. Uniform Acceleration and SUVAT Equations | 匀加速运动与 SUVAT 方程

For motion with constant acceleration, the SUVAT equations link displacement s, initial velocity u, final velocity v, acceleration a and time t.

对于加速度恒定的运动，SUVAT 方程将位移 s、初速度 u、末速度 v、加速度 a 和时间 t 联系起来。

v = u + at

s = ut + ½at²

v² = u² + 2as

s = (u + v)t / 2

These equations only apply when acceleration is constant. Identify the known quantities first, then choose the equation that does not contain the unknown you are trying to find.

这些方程只在加速度恒定时适用。先确定已知量，再选择不含待求未知量的方程。

8. Worked Example: Car Acceleration | 例题：汽车加速

A car accelerates uniformly from rest to 30 m s⁻¹ in 8.0 s. Calculate its acceleration and the distance travelled during this time.

一辆汽车从静止开始匀加速到 30 m s⁻¹，用时 8.0 s。求它的加速度和行驶的距离。

First identify the known quantities: u = 0, v = 30 m s⁻¹, t = 8.0 s. Using v = u + at:

首先确定已知量：u = 0、v = 30 m s⁻¹、t = 8.0 s。利用 v = u + at：

a = (v − u) / t = (30 − 0) / 8.0 = 3.75 m s⁻²

Then calculate the displacement using s = ut + ½at²:

然后用 s = ut + ½at² 计算位移：

s = 0 + ½ × 3.75 × (8.0)² = 120 m

You can also use the average velocity method: s = (u + v)t / 2 = (0 + 30) × 8.0 / 2 = 120 m.

也可以用平均速度法：s = (u + v)t / 2 = (0 + 30) × 8.0 / 2 = 120 m。

9. Acceleration Due to Gravity | 重力加速度

Near the Earth’s surface, all objects in free fall have the same downward acceleration g, approximately 9.81 m s⁻², assuming air resistance is negligible.

在地球表面附近，忽略空气阻力时，所有自由落体都具有相同的向下加速度 g，约为 9.81 m s⁻²。

When applying SUVAT equations to vertical motion, choose a sign convention first. If upward is taken as positive, then the acceleration due to gravity is a = −9.81 m s⁻².

将 SUVAT 方程用于竖直运动时，先规定正方向。若取向上为正，则重力加速度为 a = −9.81 m s⁻²。

A ball thrown vertically upwards has zero velocity at its highest point, but its acceleration is still −9.81 m s⁻² at that instant.

竖直上抛的小球在最高点速度为零，但该时刻其加速度仍为 −9.81 m s⁻²。

10. Measuring Acceleration in the Laboratory | 实验室测量加速度

Acceleration can be measured using a motion sensor, light gates or a ticker timer. For a trolley on a sloping track, record the velocity at two positions and the time taken to travel between them.

加速度可用运动传感器、光电门或打点计时器来测量。对于斜面上的小车，记录两个位置的速度以及经过两点之间的时间。

A single light gate with a double-interrupt card can give the initial and final velocities, while two light gates can give the average acceleration directly using a = (v − u) / t.

单个光电门配合双遮光片可获得初速度和末速度，而两个光电门可直接用 a = (v − u) / t 求出平均加速度。

In ticker timer experiments, the acceleration can be found from the increasing spacing of dots on the tape. The average velocity over a short interval is calculated first, then the change in velocity is divided by time.

在打点计时器实验中，可通过纸带上点间距的增大来求加速度。先计算短时间间隔内的平均速度，再用速度变化量除以时间。

11. Common Mistakes and Sign Conventions | 常见错误与符号约定

Common mistakes include using speed instead of velocity, mixing units such as km/h with seconds, and forgetting that acceleration can be negative.

常见错误包括用速率代替速度、单位混用（如 km/h 与秒一起使用）、忘记加速度可以为负。

Always convert km/h to m/s by dividing by 3.6 before using SUVAT equations. For example, 72 km/h = 20 m/s.

在使用 SUVAT 方程前，要先将 km/h 除以 3.6 转换为 m/s。例如 72 km/h = 20 m/s。

When an object slows down while moving in the positive direction, its acceleration is negative. This does not mean the object is moving backwards; it simply means the velocity is decreasing.

当物体沿正方向运动且速度减小时，其加速度为负。这并不意味着物体在向后运动，只是表示速度在减小。

12. Summary and Exam Tips | 总结与考试技巧

To calculate acceleration reliably, first identify whether the motion is uniform or non-uniform. If uniform, use a = Δv / Δt or the SUVAT equations. If non-uniform, use the gradient of a velocity-time graph or the tangent at a point.

要可靠地计算加速度，首先要判断运动是匀变速还是非匀变速。匀变速用 a = Δv / Δt 或 SUVAT 方程；非匀变速则利用速度-时间图像的斜率或某点切线斜率。

Always state units, give a direction for vector answers, and quote the final answer to the correct number of significant figures. In calculations, show the equation first, substitute values with units, and then give the final answer.

答题时务必写明单位，给出矢量答案的方向，并按正确的有效数字写出最终结果。计算题要先写公式，代入带单位的数值，再写出最终答案。

Practise rearranging equations and converting units quickly, because these skills are frequently tested in CIE A-Level Physics papers.

多练习公式变形和单位换算，因为这些技能在 CIE A-Level 物理考试中经常考查。

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Calculating the acceleration | 加速度的计算