运动、力与能量

Length and volume

Use a ruler 直尺 to measure length. Read the scale with your eye straight in front of the mark to avoid a reading error.

Use a measuring cylinder 量筒 to measure the volume of a liquid. Read the scale at the bottom of the curved surface (the meniscus 弯月面), with your eye level with it.

Measuring time

Use a stopwatch 秒表 or a digital timer 数字计时器 to measure a time interval. A clock is fine for long times (minutes or hours). Starting and stopping by hand introduces reaction time 反应时间 error; its size depends on the operator and method. To make this error matter less, time a long interval, or time many repeats and divide (see below).

Measuring small amounts

A single small length or a short time is hard to measure well. The trick is to measure many and divide 测多个再相除:

• To find the average thickness of one sheet, measure a stack of 100 sheets and divide by 100. Count sheets, not printed page numbers.
• To find the time for one swing of a pendulum 摆, measure the time for 20 swings and divide by 20. The time for one complete oscillation is called the period 周期.

This reduces the effect of a fixed reading or timing uncertainty 不确定度 on the average value. It does not remove systematic errors or a miscount.

Worked measurement comparison. A stack of 100 sheets without covers is $10.0\ \text{mm}$ thick. Count the sheets, align the stack, avoid gaps or uneven compression, and read at eye level. The thickness is an average, not a precise measurement of every sheet.

Three timings of 20 complete pendulum oscillations are $31.8\ \text{s}$, $32.2\ \text{s}$ and $32.0\ \text{s}$. Start and finish at a marked position with the pendulum moving in the same direction; count complete cycles.

Timing multiples reduces the effect of a similar start/stop error per oscillation; repeated trials help check consistency. Neither corrects a wrong cycle count.

Scalars and vectors

A scalar quantity 标量 has size (magnitude 大小) only. A vector quantity 矢量 has size and direction.

• Scalars: distance, speed, time, mass 质量, energy 能量, temperature 温度.
• Vectors: force 力, weight 重力, velocity 速度, acceleration 加速度, momentum 动量, electric field strength 电场强度, gravitational field strength 重力场强度.

To add two vectors at right angles (90°), draw them as two sides of a rectangle. The resultant 合矢量 is the diagonal. Find its size with Pythagoras and its direction with trigonometry.

Speed and velocity

Speed 速率 is the distance travelled per unit time.

Velocity is speed in a stated direction. So velocity is a vector but speed is a scalar.

Worked example. A cyclist rides $3.6\ \text{km}$ to school. The ride takes $10$ minutes, plus $2$ minutes waiting at traffic lights. Find the average speed in m/s.

Change to SI units first: $3.6\ \text{km} = 3600\ \text{m}$ and $12\ \text{min} = 12 \times 60 = 720\ \text{s}$. The waiting time counts, because it is part of the total time.

Acceleration

Acceleration is the change in velocity per unit time.

Here $\Delta v$ means "the change in velocity". A deceleration 减速 reduces speed: the acceleration is opposite to the velocity. Its sign depends on the chosen positive direction.

Worked example. A car speeds up from $8\ \text{m/s}$ to $20\ \text{m/s}$ in $4.0\ \text{s}$. Find its acceleration.

Motion graphs

A distance–time graph 距离-时间图 shows how far an object has gone:

• A flat (horizontal) line means the object is at rest 静止.
• A straight slope means constant speed. The gradient 斜率 (steepness) is the speed.
• A curve that gets steeper means the object is speeding up.

A speed–time graph 速度-时间图 shows how fast an object is going:

• A flat line means constant speed.
• A straight slope means constant acceleration. The gradient is the acceleration.
• The area under the line 线下面积 is the distance travelled.

Worked example (reading a graph). The graph shows a bus moving between two stops. Find (a) the acceleration in the first $6\ \text{s}$, (b) the acceleration in the last $4\ \text{s}$, (c) the distance between the stops and (d) the average speed.

(a) Gradient of the first part: $a = \dfrac{12 - 0}{6 - 0} = 2.0\ \text{m/s}^2$.

(b) Gradient of the last part: $a = \dfrac{0 - 12}{20 - 16} = -3.0\ \text{m/s}^2$. The minus sign shows the bus is slowing down: a deceleration of $3.0\ \text{m/s}^2$.

(c) Distance = total area under the line:

(d) Average speed $= \dfrac{180\ \text{m}}{20\ \text{s}} = 9.0\ \text{m/s}$. This is less than the top speed of $12\ \text{m/s}$, because the bus is slower while it speeds up and slows down.

Read the time for each part from the time axis: part B lasts $16 - 6 = 10\ \text{s}$, not $16\ \text{s}$.

Falling objects

Near the Earth the gravitational field is uniform 均匀 (the same strength everywhere), so objects in free fall have the same acceleration when air resistance is negligible. This is the acceleration of free fall 自由落体加速度, $g \approx 9.8\ \text{m/s}^2$.

When an object falls through air, air resistance 空气阻力 (a drag force) acts upward. As it speeds up, air resistance grows. When air resistance equals the weight, the resultant force is zero and the object stops speeding up. It then falls at a steady terminal velocity 收尾速度.

Worked example. A skydiver of mass $70\ \text{kg}$ falls through the air ($g = 10\ \text{N/kg}$, so the weight is $700\ \text{N}$).

• When the air resistance is $500\ \text{N}$: resultant force $= 700 - 500 = 200\ \text{N}$ downwards, so $a = \dfrac{F}{m} = \dfrac{200}{70} = 2.9\ \text{m/s}^2$ downwards. The skydiver is still speeding up, but less quickly than at the start.
• At terminal velocity the air resistance is $700\ \text{N}$, equal to the weight. Resultant force $= 0$, so $a = 0$.
• The parachute opens and the air resistance jumps to $1200\ \text{N}$. Resultant force $= 1200 - 700 = 500\ \text{N}$ upwards, so $a = \dfrac{500}{70} = 7.1\ \text{m/s}^2$ upwards. The skydiver is still moving down but slows down, until the air resistance falls back to $700\ \text{N}$ at a new, lower terminal velocity.

An upward resultant force does not make the skydiver go up. It makes the downward speed get smaller.

Mass is the amount of matter 物质 in an object. It is measured in kilograms (kg) and does not change when you move the object.

Weight is the force of gravity on a mass. It is measured in newtons (N). Weight can change: it is smaller on the Moon because the Moon's gravity is weaker.

Gravitational field strength is the force per unit mass:

This $g$ has the same value as the acceleration of free fall ($\approx 9.8\ \text{N/kg}$).

Worked example. An astronaut has a mass of $60\ \text{kg}$. Find the weight on the Earth ($g = 9.8\ \text{N/kg}$) and on the Moon ($g = 1.6\ \text{N/kg}$).

The mass is $60\ \text{kg}$ in both places. Only the weight changes. You can compare masses with a balance 天平.

Density is the mass per unit volume.

The symbol $\rho$ is the Greek letter "rho". The unit is $\text{kg/m}^3$ or $\text{g/cm}^3$.

To find density: measure the mass with a balance, find the volume, then divide.

• Regular solid 规则固体 (like a box): measure the sides and calculate the volume.
• Irregular solid 不规则固体 (a strange shape): lower it into water in a measuring cylinder. The rise in water level is its volume. This is the displacement method 排水法.
• Liquid: put an empty measuring cylinder on the balance and read its mass. Pour in the liquid, read the volume on the cylinder, and read the mass again. The mass of the liquid is the difference between the two mass readings.

An object floats 漂浮 if its density is less than the density of the liquid. It sinks if its density is greater. In the same way, one liquid floats on top of another (if the two do not mix) when its density is lower: oil floats on water.

Worked example. A stone of mass $54\ \text{g}$ is lowered into a measuring cylinder. The water level rises from $20\ \text{cm}^3$ to $40\ \text{cm}^3$. Find the density of the stone.

The volume of the stone is the rise in water level, $40 - 20 = 20\ \text{cm}^3$, so

A force is a push or a pull. A force can change the shape 形状, the speed, or the direction of an object.

Stretching (Hooke's law)

When you hang a load on a spring, it stretches. The stretch is called the extension 伸长量.

On a load–extension graph 载荷-伸长图 the line is straight at first: extension is proportional to load. The point where the line stops being straight is the limit of proportionality 比例极限.

The experiment. Hang the spring from a clamp and stand a ruler beside it. Read the position of the bottom of the spring with no load: this is the original length. Add loads one at a time (for example 1 N each) and record the new length each time. The extension is the new length minus the original length. Plot load on the y-axis against extension on the x-axis.

Within the linear range, the spring constant 弹簧常数 is force divided by extension:

A large $k$ means a stiff spring.

Worked example. A spring is $12.0\ \text{cm}$ long with no load. With a $6.0\ \text{N}$ load it is $15.0\ \text{cm}$ long. Find (a) the spring constant and (b) the length with a $4.0\ \text{N}$ load (the limit of proportionality is not reached).

(a) Extension $x = 15.0 - 12.0 = 3.0\ \text{cm} = 0.030\ \text{m}$, so

(b) Extension $= \dfrac{F}{k} = \dfrac{4.0}{200} = 0.020\ \text{m} = 2.0\ \text{cm}$, so the length is $12.0 + 2.0 = 14.0\ \text{cm}$.

Use the extension in $k = F/x$, never the whole length: $6.0 \div 0.150$ gives a wrong answer.

Resultant force and Newton's laws

Add forces on a straight line to get the resultant force 合力 (forces one way are positive, the other way negative).

• If the resultant force is zero, the object stays at rest or keeps moving in a straight line at constant speed. (Newton's first law.)
• If the resultant force is not zero, the object accelerates in the direction of the force:

Worked example. A $1200\ \text{kg}$ car has a $3000\ \text{N}$ driving force and $600\ \text{N}$ of friction. Find its acceleration.

First find the resultant force: $3000 - 600 = 2400\ \text{N}$. Then

Motion in a circle

A resultant force at right angles to the motion does not change the speed. It changes the direction, so the object moves in a circle at a steady speed. This is circular motion 圆周运动. The force always points towards the centre of the circle: the pull of the string on a whirling ball, or the gravity of the Sun on a planet.

For an object moving in a circle:

• a bigger force is needed to keep the same mass moving at a higher speed in a circle of the same radius 半径;
• a bigger force makes the same mass at the same speed move in a smaller circle (a smaller radius);
• a bigger mass needs a bigger force to keep the same speed and the same radius.

Friction

Friction 摩擦力 is the force between two surfaces that touch. It opposes relative sliding or the tendency to slide, and can cause heating. Drag opposes an object's motion relative to the surrounding fluid.

Moments — the turning effect

The moment 力矩 of a force is its turning effect about a pivot 支点.

The unit is the newton metre (N m).

Principle of moments 力矩原理: when an object is balanced (in equilibrium 平衡),

An object is in equilibrium when there is no resultant force and no resultant moment.

Worked example. A $30\ \text{N}$ weight sits $0.20\ \text{m}$ to the left of a pivot. How far to the right must a $20\ \text{N}$ weight sit to balance the beam?

Balanced, so anticlockwise moment $=$ clockwise moment:

Checking the principle by experiment. Balance a metre rule at its centre on a pivot. Hang a known weight on each side and move them until the rule balances again. Measure each distance from the pivot. Force × distance on the left equals force × distance on the right, within the small errors of measurement, so there is no resultant moment on a balanced object.

Centre of gravity

The centre of gravity 重心 is the single point where all the weight of an object seems to act.

For a flat shape (lamina 薄片), hang it from a pin and let it settle; draw a vertical line down from the pin using a plumb line. Repeat from another point. The centre of gravity is where the lines cross.

The stability 稳定性 of an object depends on where its centre of gravity is. An object is more stable 稳定 when its centre of gravity is low and its base is wide. It tips over when a vertical line down from the centre of gravity passes outside the base.

Momentum is mass times velocity. It is a vector.

Impulse 冲量 is the force times the time it acts, and it equals the change in momentum:

So the resultant force is the change in momentum per unit time:

Worked example. A $0.40\ \text{kg}$ ball hits a wall at $10\ \text{m/s}$ and bounces straight back at $8.0\ \text{m/s}$. It touches the wall for $0.050\ \text{s}$. Find the impulse and the average force on the ball.

Take towards the wall as positive, so the velocity after is $-8.0\ \text{m/s}$.

The impulse is $7.2\ \text{N s}$ away from the wall, and

away from the wall. If you forget the change of direction you get $0.40 \times (10 - 8.0) = 0.80\ \text{N s}$, which is far too small.

Conservation of momentum 动量守恒: when objects collide and no outside force acts, the total momentum before equals the total momentum after.

(Here $u$ is a velocity before and $v$ is a velocity after.)

Worked example. A $2.0\ \text{kg}$ trolley moving at $3.0\ \text{m/s}$ hits a stationary $1.0\ \text{kg}$ trolley and they stick together. Find their common velocity afterwards.

Total momentum is conserved, so

Energy stores

Energy can be stored 储存 in different ways: kinetic 动能, gravitational potential 重力势能, chemical 化学能, elastic (strain) 弹性势能, nuclear 核能, electrostatic 静电能, and internal (thermal) 内能.

Energy is transferred 转移 between stores by forces (mechanical work), by electric currents, by heating, and by waves (such as light and sound).

Kinetic and potential energy

Kinetic energy is the energy of a moving object:

The change in gravitational potential energy when an object goes up or down by a height $\Delta h$:

Conservation of energy

The principle of conservation of energy 能量守恒定律 says energy is never made or destroyed; it only moves between stores. A falling object turns gravitational potential energy into kinetic energy. A Sankey diagram 桑基图 shows how the input energy splits into useful energy and wasted 浪费 energy.

Worked example. A $0.50\ \text{kg}$ ball is dropped from a height of $1.8\ \text{m}$. Ignoring air resistance, find its speed just before it lands. (Take $g = 10\ \text{m/s}^2$.)

All the gravitational potential energy becomes kinetic energy, so $\tfrac{1}{2}mv^2 = mg\,\Delta h$. The mass cancels, leaving $v^2 = 2g\,\Delta h$:

Because the mass cancels, every object dropped from this height would land at the same speed.

Work

Work done 做功 equals the energy transferred. For a constant force, use the displacement in the direction of that force:

The unit of work and energy is the joule 焦耳 (J).

The distance $d$ must be in the direction of the force. When you lift something, use the vertical height, not the length of a slope or a staircase.

Power

Power 功率 is the work done (or energy transferred) per unit time.

The unit is the watt 瓦特 (W). $1\ \text{W} = 1\ \text{J/s}$.

Worked example. A student of weight $500\ \text{N}$ runs up a staircase. The stairs rise $3.0\ \text{m}$ vertically and are $5.0\ \text{m}$ long along the slope. The climb takes $6.0\ \text{s}$. Find the work done against gravity and the power.

The force (the weight) acts vertically, so use the vertical height $3.0\ \text{m}$:

A student who walks up the same stairs in $15\ \text{s}$ does the same work, $1500\ \text{J}$, but with a smaller power, $100\ \text{W}$.

Efficiency

Efficiency 效率 tells you how much of the input energy becomes useful energy.

Real devices usually transfer some input energy to less useful stores, so their useful efficiency is below 100%. An ideal model can use 100% efficiency; wasted energy is transferred, not destroyed. The same idea works with power:

Worked example. A motor is supplied with $200\ \text{J}$ of electrical energy and lifts a load, giving it $150\ \text{J}$ of gravitational potential energy. Find its efficiency.

Comparing energy and power. Suppose the same motor transfers this energy in $5.0\ \text{s}$. Use the same interval for both average powers.

The power and energy ratios agree because the same time interval applies. Wasted energy is transferred to less useful stores, not destroyed.

Energy resources

We generate electricity from many energy resources. Most spin a turbine that drives a generator – often by boiling water to make steam (fossil fuels, nuclear fuel, geothermal, biofuels), or directly by moving water or air (hydroelectric, waves, tides, wind). Solar cells make electricity from sunlight directly; solar panels heat water.

The nuclear reactor does not emit carbon dioxide while operating, but fuel supply and construction can cause emissions. Biofuel crops can absorb carbon dioxide as they regrow; emissions from growing, processing and transporting the fuel also matter, so replanting alone does not guarantee zero net emissions.

When you compare resources, use these five ideas: renewability 可再生性 (will it run out?), availability 可用性 (is it always there? the wind can drop and the Sun sets), reliability 可靠性 (does it give a steady output?), scale 规模 (how much energy can it supply?) and environmental impact 环境影响 (pollution, carbon dioxide, land flooded by a dam).

Most of these trace back to the Sun (fossil fuels are ancient stored sunlight; wind and waves come from solar heating) – the exceptions are geothermal, nuclear, and tidal energy. The Sun itself is powered by nuclear fusion 核聚变, joining hydrogen nuclei into helium. The syllabus includes research into using energy released by fusion to generate electrical power on a large scale.

Worked energy routes. A school wants electric lighting and hot water. Solar cells supply electrical energy directly; solar water-heating panels transfer energy from solar radiation to the internal energy of water. Neither needs a steam turbine for its stated function. In a fuel-fired power station, a boiler heats water, steam turns a turbine and the turbine drives a generator. Wind or moving water can turn turbines directly. Identify the energy route before naming the components.

Exam transfer. In 0625/22 May/June 2025 Q9, wind is the only one of wind, geothermal and nuclear fuel that gets its energy from the Sun (official answer D). Solar heating helps drive movement of air. Geothermal energy comes from inside the Earth; nuclear fuel supplies energy through nuclear changes. Renewability is a separate test.

Worked resource comparison. A remote clinic needs electricity day and night. Solar cells without a battery, backup or grid connection cannot meet its night demand. A hydroelectric scheme may supply at night if suitable flow or stored water is available, but seasonal flow and scheme capacity must be checked. Compare each supply with the clinic's power requirement and assess site, land and habitat effects, especially if a dam is needed. Renewable alone does not mean continuous, adequate or environmentally harmless. Use availability, reliability, scale and environmental impact to explain the choice.

Pressure is the perpendicular force per unit surface area.

The unit is the pascal 帕斯卡 (Pa). $1\ \text{Pa} = 1\ \text{N/m}^2$.

A small area gives a large pressure (a sharp knife cuts well). A large area gives a small pressure (snowshoes stop you sinking).

Worked example. A $600\ \text{N}$ student stands on both feet. Each shoe touches the floor over $200\ \text{cm}^2$. Find the pressure on the floor.

The total area is $400\ \text{cm}^2$. Change it to m²: $1\ \text{m}^2 = 100 \times 100 = 10\,000\ \text{cm}^2$, so $400\ \text{cm}^2 = 0.040\ \text{m}^2$.

Standing on one foot halves the area, so the pressure doubles to $30\,000\ \text{Pa}$.

Pressure in a liquid

In a liquid, pressure increases with depth 深度 and with the liquid's density:

This is why a dam is built thicker at the bottom, where the water pressure is greatest. Pressure in a liquid acts in all directions.

Worked example. Find the extra pressure at a depth of $2.0\ \text{m}$ in water (density $1000\ \text{kg/m}^3$, $g = 10\ \text{N/kg}$).

• On a distance–time graph the gradient is the speed; on a speed–time graph the gradient is the acceleration and the area under the line is the distance travelled. Never read one graph as if it were the other.
• Mass (kg) is the amount of matter and is the same everywhere; weight (N) is the force of gravity, $W = mg$. On the Moon your mass is unchanged but your weight is smaller.
• Speed is a scalar; velocity is a vector. Something moving at steady speed around a curve is still accelerating, because its direction keeps changing.
• In Hooke's law, load is proportional to extension only up to the limit of proportionality. Use the extension (stretched length − original length), never the whole length.
• Total momentum is conserved in a collision when the resultant external force is negligible. Give each velocity a $+$ or $-$ sign for its direction before you add them.
• Check that units are consistent. $p=F/A$ defines pressure; $\Delta p=\rho g\,\Delta h$ gives the change in pressure between depths in a liquid.

Common mistakes

• Averaging two speeds to get the average speed. Use total distance ÷ total time, and include the time spent stopped.
• Saying "no forces act" at terminal velocity. The weight and the air resistance both act; they are equal and opposite, so the resultant force is zero.
• Writing a weight in kilograms. Weight is a force, measured in newtons; mass is in kilograms.
• Using the whole length of a spring in $k = F/x$. Use the extension (new length − original length), in metres.
• Forgetting the minus sign when an object bounces back. The change in velocity from $+10$ to $-8\ \text{m/s}$ is $-18\ \text{m/s}$, not $-2\ \text{m/s}$.
• Dividing by an area in cm² in $p = F/A$. Change to m² first: $1\ \text{cm}^2 = 0.0001\ \text{m}^2$.
• Using the slope length to find the work done in lifting. Use the vertical height, because the weight acts vertically.