Position and Movement (I)
FoundationDescribing motion starts with distance versus displacement and speed versus velocity: displacement and velocity are vectors with direction. On a displacement-time graph the slope is velocity; on a velocity-time graph the slope is acceleration and the area under it is displacement.
- State the reference first; distance and speed are scalars, displacement and velocity are vectors
- Average velocity = displacement ÷ time; average speed = distance ÷ time
- s-t graph slope = velocity: horizontal = at rest, straight = uniform motion, curve = accelerating or decelerating
文體小知識：科學知識說明文——講解路程與位移、速率與速度、標量與向量，並訓練位移—時間圖與速度—時間圖的讀圖能力。
§Text Passage
1Mechanics begins with describing motion. Before studying motion, choose a reference and describe how the object's position changes. Distance is the total length of the path actually travelled and has only magnitude; displacement points straight from the start to the end and carries direction, so the two need not agree. For example, walk 100 m forward along a straight road and then 40 m back: the distance covered is 140 m but the displacement is only 60 m forward. Telling the two apart is the first step of kinematics.
2Physical quantities divide into scalars and vectors: scalars have only magnitude — distance, speed, time; vectors have magnitude and direction — displacement, velocity, acceleration. Velocity is the rate of change of displacement: average velocity equals displacement divided by time, while average speed equals distance divided by time. Direction changes the result. For example, run a full lap of the track back to the start: the distance is several hundred metres and the average speed is not zero, but the displacement is zero and so is the average velocity.
3Instantaneous and average values must also be distinguished: a car's speedometer shows instantaneous speed — how fast at that moment — while average velocity is the total displacement over the whole time. The best way to present motion is a graph: the displacement-time (s-t) graph and the velocity-time (v-t) graph. On an s-t graph the slope is the velocity: the steeper the line, the faster the motion; a zero slope means the object is at rest.
4Acceleration describes how quickly velocity changes, defined as the rate of change of velocity, in metres per second per second. Speeding up is acceleration; slowing down involves acceleration too — just directed opposite to the motion. For example, a train gaining 2 m s⁻¹ of speed every second has an acceleration of 2 m s⁻²; before stopping, losing 2 m s⁻¹ each second is a deceleration of the same size, opposite to the motion.
5A v-t graph is equally rich: its slope gives the acceleration, and the area between the line and the time axis gives the displacement. Uniform motion appears as a horizontal line whose area equals velocity times time. For example, walking at 10 m s⁻¹ for 5 s gives a horizontal line at height 10, and the area 10 × 5 = 50 m is exactly the displacement. Once you can read these two graphs, any straight-line motion can be 'translated'.
1Summary 重點摘要
Describing motion starts with distance versus displacement and speed versus velocity: displacement and velocity are vectors with direction. On a displacement-time graph the slope is velocity; on a velocity-time graph the slope is acceleration and the area under it is displacement.
2Vocabulary cards Vocabulary
The location of an object relative to a chosen reference; a reference must be chosen before describing motion.
The total length of the path actually travelled; a scalar.
The change of position pointing straight from start to end, with magnitude and direction; a vector.
A quantity with magnitude only, such as distance, speed, time and mass.
A quantity with both magnitude and direction, such as displacement, velocity, acceleration and force.
Distance divided by time; magnitude only. Instantaneous speed is how fast at a particular moment.
Displacement divided by time, with magnitude and direction; average velocity is total displacement over total time.
The rate of change of velocity, in m s⁻²; 'accelerating' covers speeding up, slowing down and changing direction.
A graph of displacement against time; its slope gives the velocity.
A graph of velocity against time; its slope gives the acceleration and the area under it gives the displacement.
3Step by step Steps
- Step 1: Fix the reference firstBefore any talk of motion, state what the motion is relative to. Position is an object's location relative to a reference, and different references give different descriptions. For example, sitting in a moving train you are at rest relative to the carriage yet moving fast relative to the ground — both statements are correct physics, they just use different references. DSE questions generally take the ground as the reference unless stated otherwise. Get into the habit of stating the reference first; you will need the same habit when forces arrive.
- Step 2: Distance and displacementDistance and displacement are easily confused. Distance is the total length of the path travelled and is never negative; displacement is an arrow pointing straight from start to end, with size and direction, fixed by the two ends and independent of the path. For example, walk 100 m forward and then 40 m back: distance 140 m, displacement 60 m forward; walk 100 m back to the start: distance 200 m, displacement zero. When a question asks for displacement, always state the direction (forward or backward) — magnitude without direction loses marks.
- Step 3: Scalars and vectorsQuantities divide into scalars and vectors: scalars have magnitude only, vectors magnitude plus direction. In this topic, distance, speed, time and mass are scalars; displacement, velocity, acceleration — and soon force — are vectors. Vectors must be added with their directions in mind. For example, two people walking at 3 m s⁻¹, one north and one south, have the same speed but different velocities — direction is the whole difference. In straight-line motion we often take the forward direction as positive, so a southward 5 m s⁻¹ can be written as −5 m s⁻¹; the minus sign carries the direction.
- Step 4: Speed and velocitySpeed and velocity also need separating. Average speed is distance over time; average velocity is displacement over time, in the direction of the displacement. On a round trip the displacement can be zero, so the average velocity is zero while the average speed is not. For example, run a lap of a circular track of radius 50 m at a steady 4 m s⁻¹: the lap takes about 78.5 s, the distance is 314 m, the average speed is 314 ÷ 78.5 = 4 m s⁻¹ — but the displacement is zero and so is the average velocity. Instantaneous speed, shown on a car speedometer, is how fast at one instant.
- Step 5: Reading s-t graphsOn a displacement-time graph the slope is the velocity: steeper means faster; a zero slope (horizontal line) means at rest; a negative slope means moving backward; a curve whose slope keeps changing means accelerating or decelerating. For example, a train leaving a station gives an s-t graph that curves ever steeper at first (accelerating), straightens into a line (uniform motion), then flattens to zero slope as it brakes into the next station. Read the axes and units first, then narrate segment by segment: '0–20 s accelerating, 20–60 s uniform, 60–75 s decelerating to a stop'.
- Step 6: Reading v-t graphsA velocity-time graph gives two readings: the slope is the acceleration and the area between the line and the time axis is the displacement. A horizontal line means uniform motion; an upward sloping line means uniform acceleration; a downward one means uniform deceleration; where the line crosses the axis into negative values the object moves backward. For example, a train cruising at 20 m s⁻¹ for 40 s draws a horizontal line at height 20 whose area, 20 × 40 = 800 m, is the displacement; if it accelerates from 0 to 20 m s⁻¹ in 20 s, the slope is 1 and the acceleration is 1 m s⁻².
- Step 7: The two graphs — one motion, two viewsThe same motion can be drawn as an s-t graph and a v-t graph, and the two correspond: a straight segment on the s-t graph (fixed slope) matches a horizontal segment on the v-t graph (uniform velocity); a curve on the s-t graph matches a sloping line on the v-t graph (accelerating or decelerating); a horizontal s-t segment matches a v-t line on the time axis (at rest). For example, when a train stands at a station its s-t graph is horizontal and its v-t graph hugs the axis; as it starts, the s-t graph bends upward while the v-t line rises from zero. Identify the axes first, then compare segment by segment.
4Diagrams Diagrams
5Worked example Example
Example: an MTR train runs from station A to station B. From 0 to 20 s it accelerates uniformly from rest to 20 m s⁻¹; from 20 to 60 s it travels at 20 m s⁻¹; from 60 to 75 s it decelerates uniformly and stops. Find (a) the acceleration during the first stage, (b) the total displacement, and (c) in which stage it is fastest and where it stops.
- First stage: a = (v − u)/t = (20 − 0)/20 = 1 m s⁻²
- First-stage displacement: s₁ = ½ × 20 × 20 = 200 m (area of the triangle on the v-t graph)
- Cruising displacement: s₂ = 20 × 40 = 800 m (rectangle); braking displacement: s₃ = ½ × 20 × 15 = 150 m
- Total displacement: s = 200 + 800 + 150 = 1150 m; fastest stage is the uniform cruise at 20 m s⁻¹; in the last 15 s (60–75 s) it decelerates to rest at station B
- Answer: (a) Acceleration 1 m s⁻²; (b) total displacement 1150 m; (c) fastest is the uniform stage (20 m s⁻¹) between 20 and 60 s, and between 60 and 75 s it decelerates to rest at station B. The whole v-t graph is a triangle, a rectangle and a reversed triangle; the sum of their areas is the total displacement.
6Common mistakes Mistakes
7Quick check Quick check
Q1What is the difference between displacement and distance? Give one example.
Q2Which of these are vectors: time, displacement, speed, acceleration?
Q3What do the slopes of an s-t graph and a v-t graph represent?
Q4How is displacement found from a v-t graph?
Q5A train travels at 15 m s⁻¹ for 30 s. What is its displacement?
8Recap Recap
- State the reference first; distance and speed are scalars, displacement and velocity are vectors
- Average velocity = displacement ÷ time; average speed = distance ÷ time
- s-t graph slope = velocity: horizontal = at rest, straight = uniform motion, curve = accelerating or decelerating
- v-t graph slope = acceleration; area under the graph = displacement
- Match the two graphs: straight s-t ↔ horizontal v-t; curved s-t ↔ sloping v-t