Chapter 2 · Section 6Textbook pages 40–44

Position–Time and Velocity–Time Graphs ചലനത്തിന്റെ ഗ്രാഫിക ചിത്രീകരണം

A table of numbers tells you what happened; a graph lets you see it at a glance. On this page you draw a position–time graph point by point, read a car's velocity from the slope of its graph, and discover that in a velocity–time graph the area under the line is the displacement.

📖 What you need (pages 3, 4 and 6 of this guide)

Plotting a point (x, y) and choosing a scale – page 3 · Drawing & reading graphs.
Area of a rectangle (length × breadth) and a triangle (½ × base × height) – page 4.
Velocity v = s/t (displacement ÷ time), unit m/s – page 6; acceleration a = (v − u)/t, unit m/s² – page 8.

1. What is a graph? – Graph 2.2

ഗ്രാഫ് ഒരു ദ്വിമാനചിത്രമാണ്. A graph is a two-dimensional picture. The horizontal line XX′ is the X-axis (X അക്ഷം) and the vertical line YY′ is the Y-axis (Y അക്ഷം). They cross at the origin O (മൂലബിന്ദു). OX (to the right) is positive, OX′ (to the left) is negative; OY (up) is positive and OY′ (down) is negative. Both axes are number lines.

Graph paper is ruled in squares 1 cm long and 1 cm wide (with thin millimetre lines inside). The square ABCD has an area (പരപ്പളവ്) of 1 cm². A graph is named after the quantities on its axes in the order Y-quantity – X-quantity: position on Y and time on X gives a “position–time graph”.

🧪 Explore the axes (drag the point or use the sliders)

x cm y cm

Why draw graphs?

ഗ്രാഫ് ഉപയോഗിച്ച് അളവുകൾ തമ്മിലുള്ള ബന്ധം മനസ്സിലാക്കാനും ചിത്രീകരിക്കാനും ഈ ബന്ധത്തെ അടിസ്ഥാനപ്പെടുത്തി സമവാക്യം രൂപീകരിക്കാനും കഴിയും.

🔗 See relations between two quantities

✏️ Form equations from that relation

🧮 Do calculations (slope, area)

⚡ Get instant information without a table

💡 Draw conclusions (trends, highs and lows)

The road-accident graph (Graph 2.1) and its questions – which year had the fewest accidents, how many in 2019 – are on page 9.

🔬 ExpEYES → Distance measurement using SR04 echo module, plotting graphs

ExpEYES (Experiments for Young Engineers and Scientists) is a low-cost, open-source lab kit used in many Kerala schools; it runs on free software, so you need no special licence. Its SR04 ultrasonic echo module sends out a short burst of sound and times the echo from an object: distance = (speed of sound × echo time) ÷ 2. Measuring this many times a second, ExpEYES plots a position–time graph live on the computer screen as you move a book towards or away from the sensor.

Try it in your school lab: walk the book away steadily – you get a sloping straight line (uniform velocity). Stop – the line becomes horizontal. Move it faster – the line gets steeper.

2. Position–time graph – Table 2.8 and Graph 2.3

ഒരു വസ്തുവിന്റെ ചലനവുമായി ബന്ധപ്പെട്ട വിവരങ്ങളാണ് പട്ടികയിൽ നല്കിയിരിക്കുന്നത്. The table gives the position of an object at different times. Draw the axes X′OX and Y′OY, choose a suitable scale (തോത്), put time on the X-axis and position on the Y-axis, and mark each ordered pair as a point. For (2, 1): go up from 2 on the X-axis and across from 1 on the Y-axis; where they meet is the point.

X-axis – time (s)	2	4	6	8	10
Y-axis – position (m)	1	2	3	4	5

Table 2.8 · Scale in the textbook (Graph 2.3): X-axis 1 cm = 2 s, Y-axis 1 cm = 1 m

🧪 Graph plotter – plot Table 2.8 and try other scales

Scale

Change the scale after joining the points: the line becomes steeper or flatter on the paper, but every scale gives the same velocity, 0.5 m/s.

Questions on Graph 2.3 (page 42)

Q1. ലഭിച്ച ഗ്രാഫിന്റെ സ്വഭാവമെന്ത്? (തിരശ്ചീനമായ നേർരേഖയാണ്/ചരിവുള്ള നേർരേഖയാണ്/വക്രരേഖയാണ്) – What is the shape of the graph: a horizontal straight line, a sloping straight line or a curve?

A sloping straight line (ചരിവുള്ള നേർരേഖ), and extended back it passes through the origin.

Why: every point obeys position = 0.5 × time, so the points lie exactly on one straight line that rises 1 m for every 2 s.

Q2. ഈ ഗ്രാഫ് ഏതുപേരിൽ അറിയപ്പെടുന്നു? – What is this graph called?

A position–time graph (സ്ഥാന-സമയ ഗ്രാഫ്).

Why: a graph is named “Y-quantity – X-quantity”; here position is on the Y-axis and time on the X-axis.

Q3. ലഭിച്ച ഗ്രാഫിന്റെ ആകൃതിയിൽ നിന്ന് വസ്തുവിന്റെ പ്രവേഗത്തിന്റെ പ്രത്യേകത എന്ത്? (സമപ്രവേഗം/അസമപ്രവേഗം) – From the shape of the graph, what kind of velocity does the object have?

Uniform velocity (സമപ്രവേഗം), 0.5 m/s.

Why: in every 2 s the position grows by the same 1 m – equal displacements in equal time intervals. A straight sloping p–t line always means uniform velocity; a curve would mean non-uniform velocity.

Q4. 5 s കൊണ്ട് വസ്തുവിനുണ്ടായ സ്ഥാനാന്തരം എത്രയാണ്? – What is the displacement of the object in 5 s?

2.5 m.

Why: go up from 5 s on the X-axis to the line, then across to the Y-axis – it meets halfway between 2 m and 3 m (press “Read: position at 5 s” in the plotter). Check: 0.5 m/s × 5 s = 2.5 m.

Q5. 1.5 m സഞ്ചരിക്കാനെടുത്ത സമയം എത്രയാണ്? – How long did it take to travel 1.5 m?

3 s.

Why: go across from 1.5 m on the Y-axis to the line, then down to the X-axis – it lands on 3 s. Check: t = s/v = 1.5 ÷ 0.5 = 3 s.

3. Velocity from a position–time graph – Graph 2.4

ഒരു കാറിന്റെ ചലനവുമായി ബന്ധപ്പെട്ട സ്ഥാന-സമയ ഗ്രാഫ് തന്നിരിക്കുന്നു. Here is the position–time graph of a car (scale: X-axis 1 cm = 2 s, Y-axis 1 cm = 10 m). Point A is at t₁ = 4 s, s₁ = 20 m; point B is at t₂ = 10 s, s₂ = 50 m. Pick any two times with the sliders – the coloured slope triangle gives the velocity.

🧪 Slope triangle on Graph 2.4

t₁ s t₂ s

പ്രവേഗം = സ്ഥാനാന്തരം / സമയം – velocity = displacement ÷ time:

v = s₂ − s₁t₂ − t₁ = 50 m − 20 m10 s − 4 s = 30 m6 s = 5 m/s

Questions on Graph 2.4 (page 43)

Q6. ഗ്രാഫിൽ A മുതൽ B വരെയുള്ള ഭാഗത്ത് കാറിന്റെ സ്ഥാനാന്തരം എത്ര? – What is the car's displacement from A to B?

s₂ − s₁ = 50 m − 20 m = 30 m.

Why: displacement is the change in position – read both positions on the Y-axis and subtract.

Q7. A മുതൽ B വരെ കാർ സഞ്ചരിക്കുന്നതിന് എടുത്ത സമയമെത്ര? – How long did the car take from A to B?

t₂ − t₁ = 10 s − 4 s = 6 s.

Why: read both times on the X-axis and subtract. The velocity is then 30 m ÷ 6 s = 5 m/s.

Q8. ഗ്രാഫിൽ കാറിന്റെ 6-ാം സെക്കന്റിലെ സ്ഥാനം എത്ര? – What is the car's position at the 6th second?

30 m.

Why: the point above 6 s on the line is level with 30 m on the Y-axis (it is one of the marked points).

Q9. ഈ കാറിന് ഏതുതരം പ്രവേഗമാണുള്ളത്? (സമപ്രവേഗം/അസമപ്രവേഗം) – What kind of velocity does the car have?

Uniform velocity (5 m/s).

Why: the graph is a sloping straight line – the car moves 10 m in every 2 s. Move the sliders: any pair of times gives the same 5 m/s.

Q10. 6 s നും 8 s നും ഇടയിൽ കാറിന്റെ പ്രവേഗം ഗ്രാഫിൽ നിന്ന് കണ്ടെത്തുക. – Find the car's velocity between 6 s and 8 s from the graph.

v = (40 m − 30 m) ÷ (8 s − 6 s) = 10 ÷ 2 = 5 m/s.

Why: at 6 s the car is at 30 m, at 8 s at 40 m. Same slope, same velocity – as expected for uniform velocity. Full animated solution: page 12.

⭐ Choosing a scale (textbook page 43)

തന്നിരിക്കുന്ന അളവുകളെ ഗ്രാഫ് പേപ്പറിൽ ഒതുങ്ങുന്ന രീതിയിൽ ചിത്രീകരിക്കുന്നതിനാണ് അനുയോജ്യമായ തോത് എടുക്കേണ്ടിവരുന്നത്. തോത് കൂടുന്നതിനനുസരിച്ച് ഗ്രാഫിന്റെ വലിപ്പം കുറഞ്ഞുവരും. എന്നാൽ ഗ്രാഫ് സൂചിപ്പിക്കുന്ന ഭൗതിക അളവിന്റെ (physical quantity) മൂല്യത്തിൽ വ്യത്യാസം സംഭവിക്കുന്നില്ല.

We choose a suitable scale so that the data fits on the graph paper. A bigger scale (more units per cm) makes the graph smaller – it changes how steep the line looks – but the value of the physical quantity it shows (here the velocity) does not change.

4. Velocity–time graph – Graph 2.5

X അക്ഷത്തിൽ സമയവും Y അക്ഷത്തിൽ പ്രവേഗവും രേഖപ്പെടുത്തി വരയ്ക്കുന്ന ഗ്രാഫാണ് പ്രവേഗ-സമയ ഗ്രാഫ്. A graph with time on the X-axis and velocity on the Y-axis is a velocity–time graph (പ്രവേഗ-സമയ ഗ്രാഫ്). The table and graph below belong to a vehicle (scale: X-axis 1 cm = 2 s, Y-axis 1 cm = 10 m/s).

time (s)	0	2	4	6	8	10
velocity (m/s)	0	20	40	40	40	40

From 4 s to 10 s the velocity stays 40 m/s, so the displacement is velocity × time = 40 m/s × 6 s. On the graph, 40 m/s is the height AB and 6 s is the width AD, so the displacement equals AB × AD – the area of rectangle ABCD, the region under BC. Choose any time window with the sliders and watch the shaded area turn into metres.

🧪 Area under the v–t graph = displacement

from s to s

s = 40 m/s × 6 s = 240 m

⭐ Displacement from a velocity–time graph (textbook page 44)

ഒരു പ്രവേഗ - സമയ ഗ്രാഫിൽ നിശ്ചിത ഇടവേളകൾക്കിടയിൽ വസ്തുവിനുണ്ടാകുന്ന സ്ഥാനാന്തരത്തിന്റെ അളവ് ആ സമയ ഇടവേളയിലെ ഗ്രാഫിന് ചുവടെയുള്ള ഭാഗത്തിന്റെ പരപ്പളവിന് തുല്യമായിരിക്കും.

In a velocity–time graph, the displacement of an object during a time interval equals the area under the graph for that interval. (Work the area out in graph units – m/s × s = m – not in cm².)

Questions on Graph 2.5 (page 44)

Q11. ലഭിച്ച ഗ്രാഫിൽ നിന്ന് നാലാമത്തേയും പത്താമത്തേയും സെക്കന്റിനിടയിലുണ്ടായ സ്ഥാനാന്തരം കണ്ടെത്തുക. – Find the displacement between the 4th and the 10th second.

s = AB × AD = 40 m/s × 6 s = 240 m (the area of rectangle ABCD, under BC).

Why: the velocity is constant (40 m/s) for 6 s, and displacement = velocity × time, which is exactly length × breadth of the rectangle.

Q12. ആദ്യത്തെ 4 s ലെ സ്ഥാനാന്തരം ഗ്രാഫിൽ നിന്നും കണ്ടെത്തൂ (ഗ്രാഫ് 2.5). – Find the displacement in the first 4 s from Graph 2.5.

Area of triangle OAB = ½ × base × height = ½ × 4 s × 40 m/s = 80 m.

Why: the area rule works even when the velocity changes; under the rising line the region is a triangle. (Average velocity 20 m/s × 4 s gives the same 80 m.) Full animated solution: page 12.

Q13. ആദ്യത്തെ 4 s കൊണ്ടുണ്ടായ പ്രവേഗമാറ്റം എത്ര? ത്വരണം എത്ര? – What is the change in velocity in the first 4 s? What is the acceleration?

Change in velocity = 40 − 0 = 40 m/s; acceleration a = (v − u)/t = 40 ÷ 4 = 10 m/s².

Why: in a v–t graph the slope (rise ÷ run) is the acceleration: the line rises 40 m/s in 4 s.

Q14. 4 s നും 10 s നും ഇടയിൽ ഈ വാഹനത്തിന്റെ ത്വരണം എത്ര? – What is the acceleration between 4 s and 10 s?

Zero (0 m/s²).

Why: the line BC is horizontal – the velocity stays 40 m/s, so the change in velocity is 0 and a = 0 ÷ 6 = 0. The vehicle moves with uniform velocity.

പ്രവേഗ - സമയ ഗ്രാഫിൽ നിന്നും സ്ഥാനാന്തരം, പ്രവേഗം, സമയം, ത്വരണം തുടങ്ങിയവ കണ്ടെത്താൻ കഴിയും. From one velocity–time graph you can find displacement (area), velocity (read it), time and acceleration (slope). Graphs can also give us equations – that is the next page.

🌴 Kerala connection

A Kochi Metro train between two stations has a v–t graph shaped like a flat-topped hill: a rising line (accelerating out of the station), a flat top (cruising) and a falling line (braking into the next station). The area under that “hill” is the distance between the stations – about a kilometre on many stretches. Buses and trains that log their speed every second give exactly this kind of graph to their engineers.

🌐 Explore online needs internet · opens in a new tab

📄 Physics Tutorial - 1D Kinematics - Position Time Graphs - The Meaning of Shape for a p-t Graph The Physics Classroom
Straight line = constant velocity, curve = changing velocity: learn to read position-time graphs.
📄 Physics Tutorial - 1D Kinematics - Velocity Time Graphs - Meaning of Shape for a v-t Graph The Physics Classroom
Slope of a velocity-time graph is the acceleration.
📄 Physics Tutorial - 1D Kinematics - Velocity Time Graphs - Determining the Area on a v-t Graph The Physics Classroom
Area under a velocity-time graph = displacement - the trick behind the equations of motion.
▶️ Velocity time graphs | Motion | Physics Class 9 | Khan Academy Khan Academy India - English (YouTube) · 7:21
Class 9 video: reading velocity-time graphs and finding acceleration from them.
▶️ Why distance is area under velocity-time line | Physics | Khan Academy Khan Academy (YouTube) · 9:25
Why the area under a velocity-time line gives the distance travelled.

5. Check yourself

🧭 Summary

Axes: X (OX +, OX′ −), Y (OY +, OY′ −), origin O; a graph is named “Y-quantity – X-quantity”.
Position–time graph a sloping straight line ⇒ uniform velocity; velocity = slope = (s₂ − s₁)/(t₂ − t₁).
Changing the scale changes the steepness on paper, not the value of the quantity.
Velocity–time graph: slope = acceleration, area under the graph = displacement; a horizontal line ⇒ a = 0.

Next: The equations of motion ▶