📚 Calculating Acceleration and Force | 加速度与力的计算
Calculating acceleration and force lies at the heart of CIE A-Level Physics. Students are expected to combine definitions, vector reasoning and Newton’s laws to solve both linear and inclined problems.
计算加速度和力是 CIE A-Level 物理的核心。学生需要综合定义、矢量分析和牛顿定律来解决直线和斜面问题。
1. Key Definitions and SI Units | 关键定义与国际单位
Acceleration is defined as the rate of change of velocity with respect to time. It is a vector quantity, so it has both magnitude and direction.
加速度定义为速度随时间的变化率。它是一个矢量，既有大小又有方向。
Force is a push or pull on an object that can change its velocity, direction or shape. The SI unit of force is the newton (N).
力是作用于物体上的推或拉，可以改变物体的速度、方向或形状。力的国际单位是牛顿 (N)。
| Quantity | Symbol | SI Unit | 中文名称 |
|---|---|---|---|
| Acceleration | a | m s⁻² | 加速度 |
| Force | F | N (kg m s⁻²) | 力 |
| Mass | m | kg | 质量 |
| Time | t | s | 时间 |
The average acceleration can be written as:
平均加速度可写为：
a = Δv ÷ Δt
where Δv is the change in velocity and Δt is the time taken.
其中 Δv 是速度变化量，Δt 是所用时间。
2. Newton’s Second Law: F = ma | 牛顿第二定律：F = ma
Newton’s second law states that the resultant force acting on an object is equal to the rate of change of its momentum. For constant mass, this simplifies to the familiar equation:
牛顿第二定律指出，作用在物体上的合力等于其动量的变化率。对于质量不变的情况，可以简化为熟悉的公式：
F = m × a
Here F is the resultant force in newtons, m is the mass in kilograms, and a is the acceleration in metres per second squared.
这里 F 是合力，单位为牛顿；m 是质量，单位为千克；a 是加速度，单位为米每二次方秒。
Because force and acceleration are vectors, they must have the same direction. The equation only uses the net force, not any individual force acting on the object.
由于力和加速度都是矢量，它们的方向必须相同。该方程只使用净力，而不是作用在物体上的某一个力。
From the equation, one newton can be derived as:
由该方程可以推导出牛顿的定义：
1 N = 1 kg × 1 m s⁻² = 1 kg m s⁻²
This means that a resultant force of 1 N causes a 1 kg mass to accelerate at 1 m s⁻².
这意味着 1 N 的合力使 1 kg 的物体产生 1 m s⁻² 的加速度。
3. Calculating Acceleration from Kinematic Data | 从运动学数据计算加速度
Acceleration can also be found from kinematic measurements without directly measuring force. The four equations of motion for uniform acceleration are:
加速度也可以从运动学测量中求得，而无需直接测量力。匀加速运动的四个运动学方程为：
v = u + at
s = ut + ½at²
v² = u² + 2as
s = (u + v)t ÷ 2
where u is initial velocity, v is final velocity, s is displacement, a is acceleration and t is time.
其中 u 是初速度，v 是末速度，s 是位移，a 是加速度，t 是时间。
Example: a car increases its velocity from 10 m s⁻¹ to 30 m s⁻¹ in 5.0 s. The acceleration is:
示例：一辆汽车在 5.0 秒内速度从 10 m s⁻¹ 增加到 30 m s⁻¹。其加速度为：
a = (30 – 10) ÷ 5.0 = 4.0 m s⁻²
The positive sign shows that the acceleration is in the same direction as the velocity increase.
正号表示加速度与速度增加的方向相同。
4. Resolving Forces and Finding Resultant Force | 力的分解与合力
Before applying F = ma, you must determine the resultant force acting on the object. For forces along the same line, subtract opposing forces.
在应用 F = ma 之前，必须先确定作用在物体上的合力。对于沿同一直线的力，应减去相反的力。
Consider a car of mass 1500 kg with a driving force of 500 N forwards and a resistive force of 200 N backwards.
考虑一辆质量为 1500 kg 的汽车，向前的驱动力为 500 N，向后的阻力为 200 N。
The resultant force is:
合力为：
F_net = 500 N – 200 N = 300 N
The acceleration is therefore:
因此加速度为：
a = F_net ÷ m = 300 ÷ 1500 = 0.20 m s⁻²
If forces act at an angle, resolve them into perpendicular components first, then find the net force in each direction.
如果力成一定角度作用，应先将它们分解为垂直分量，然后分别求出每个方向的合力。
5. Acceleration on a Horizontal Surface with Friction | 有摩擦的水平面加速度
On a rough horizontal surface, friction opposes motion. The frictional force f is related to the normal reaction R by:
在粗糙的水平面上，摩擦力阻碍运动。摩擦力 f 与法向反力 R 的关系为：
f = μR
where μ is the coefficient of friction. For a horizontal surface with no vertical acceleration, R equals the weight mg.
其中 μ 是摩擦系数。对于没有竖直加速度的水平面，R 等于重力 mg。
Example: a 10 kg box is pulled horizontally by a force of 50 N. The coefficient of friction is μ = 0.30. Take g = 9.81 m s⁻².
示例：一个 10 kg 的箱子被 50 N 的水平力拉动。摩擦系数 μ = 0.30。取 g = 9.81 m s⁻²。
The normal reaction is R = mg = 10 × 9.81 = 98.1 N. The friction is f = μR = 0.30 × 98.1 = 29.4 N.
法向反力为 R = mg = 10 × 9.81 = 98.1 N。摩擦力为 f = μR = 0.30 × 98.1 = 29.4 N。
The net horizontal force is F_net = 50 – 29.4 = 20.6 N, so:
水平合力为 F_net = 50 – 29.4 = 20.6 N，因此：
a = 20.6 ÷ 10 = 2.06 m s⁻²
Always subtract friction from the applied force before calculating acceleration.
计算加速度之前，务必从施加力中减去摩擦力。
6. Objects on Inclined Planes | 斜面上的物体
For an object on a smooth inclined plane at angle θ to the horizontal, the weight mg is resolved into two components:
对于放在与水平面成 θ 角的光滑斜面上的物体，重力 mg 可分解为两个分量：
Parallel component: mg sin θ
Perpendicular component: mg cos θ
With no friction, the acceleration down the slope is:
在无摩擦的情况下，沿斜面下滑的加速度为：
a = g sin θ
If friction is present, the net force becomes mg sin θ – μmg cos θ, so:
如果存在摩擦，合力变为 mg sin θ – μmg cos θ，因此：
a = g(sin θ – μ cos θ)
Example: θ = 30° and μ = 0.20. Taking g = 9.81 m s⁻², the acceleration is:
示例：θ = 30°，μ = 0.20。取 g = 9.81 m s⁻²，加速度为：
a = 9.81 × (sin 30
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导