Read speed off a distance-time graph
Scenario
Aapke exam paper mein ek graph bana hua hai. Koi formula ya number direct nahi diya gaya hai, par us ek teerchi line se aapko gaadi ki exact speed nikalni hai.
Ek simple drawn line ke 'jhukaav' (steepness) se hum object ki mathematical speed kaise nikal sakte hain?
Mental model
Graph ka slope hi gadi ka speedometer hai. Jitna khada (steep) pahad, utni fast speed.
Speed = Distance / Time. Graph mein, y-axis distance hai (Rise) aur x-axis time hai (Run). Toh Rise / Run karne se directly Speed mil jati hai. Graph ki line jitni zyada 'vertical' ki taraf point karegi, object utna hi fast bhag raha hai.
Explanation
Pichle topics mein humne padha tha ki ek straight line ka slope kaise nikalte hain, aur ye bhi ki distance-time graph me straight line uniform motion dikhati hai. Jab hum in dono concept ko jodte hain, toh graph reading ka sabse powerful tool banta hai: 'Slope of a distance-time graph gives speed'.
Isko mathematically samajhte hain. Slope ka formula hota hai (y2 - y1) / (x2 - x1). Distance-time graph me, y-axis par distance hota hai, toh (y2 - y1) object dvara chali gayi actual distance (s) ban jati hai. X-axis par time hota hai, toh (x2 - x1) liya gaya waqt (t) ban jata hai. Toh automatically aapka formula (Distance / Time) ban jata hai, jo ki exactly speed ka formula hai.
Graph se speed nikalne ke liye steps simple hain. Line par koi bhi do clear points mark kijiye jinki reading (x,y) coordinates easily grid par match ho rahi ho. Pehle unke distance ka difference nikal kar 'Rise' nikaliye, fir unke time ka difference nikal kar 'Run' nikaliye, aur unhe divide kar dijiye. Origin (0,0) aksar ek bohot aasan point hota hai calculation fast karne ke liye.
Board exams me kai baar do lines 'A' aur 'B' ek hi graph par bana dete hain, aur puchte hain 'which is moving faster?'. Jawab simple hai: Jis line ka slope (jhukav ya steepness) y-axis ki taraf zyada khada hai, uski speed naturally zyada hogi. Formula use kiye bina aap dekh kar bhi answer de sakte hain.
Worked example
Problem: A distance-time graph is a straight line passing through (0,0) and (5s, 25m). Calculate the speed of the object.
- Line origin se guzar rahi hai. Hum origin (0,0) aur point (5, 25) le sakte hain.
- Slope = (y2 - y1) / (x2 - x1) apply karte hain.
- Divide the difference in distance by difference in time.
Answer: 5 m/s
Problem: Two cars A and B have their distance-time graphs plotted on the same axes. Line A is steeper than line B. Which car is faster?
- Distance-time graph ka slope uski speed batata hai.
- Kyunki line A zyada steep (khadi) hai, iska slope bada hai, yaani wo kam time me zyada distance cover kar rahi hai.
Answer: Car A is faster.
Key points
- Slope = Speed: The numerical value of the slope of a distance-time graph is equal to the speed of the object.
- Calculation Method: Pick two points on the line. Divide the change in distance (y-axis) by the change in time (x-axis).
- Steepness: A steeper line (larger angle with the x-axis) indicates a higher speed.
Common mistakes
- Reading just one point blindly: Bache line ke beech me kisi point ka (y/x) nikal lete hain (jaise 10/2). Ye tabhi sahi hai agar line strictly origin (0,0) se start hui ho. Varna aapko pakka (y2-y1)/(x2-x1) karna padega.
- Flipping x and y: Formula galti se (Time / Distance) laga dene par unit s/m ho jati hai, jo wrong hai. Hamesha y-axis (Rise) ko numerator me rakhna chahiye.
Recall questions
- What physical quantity does the slope of a distance-time graph give?
- If graph line A is steeper than graph line B, which one represents higher speed?
- What is a key takeaway about read speed off a distance time graph?
Questions & answers
The distance-time graph of an object is given. How can you find the speed from it?
By calculating the slope of the graph. Take two points on the line, find the difference in distance (y-axis) and divide it by the difference in time (x-axis).
Approach: Explain the concept of slope (rise/run) mapping directly to speed (distance/time).
A distance-time graph for a vehicle shows a line passing through (2, 10) and (4, 30). What is its uniform speed?
10 m/s
Approach: Slope = (30 - 10) / (4 - 2) = 20 / 2 = 10 m/s.
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