Interpret uniform acceleration on velocity-time graph
Scenario
Aapki car red light se start hoti hai aur dheere-dheere apni speed lagaatar badhati hai jab tak wo highway ki limit tak nahi pahunch jati.
Speed badhne ke is straight, continuous pattern ko v-t graph me kaise show karte hain?
Mental model
Velocity-Time graph me pahad ki tarah chadhne wali (slanted) line ka matlab hai speed lagatar badh rahi hai = Uniform Acceleration.
Jab speedometer ki needle ek barabar rate se upar chadh rahi hoti hai, to graph par velocity ki line teerchi (slanted) form me upar jati hai. Ye distance-time graph ke uniform motion (constant speed) ki tarah dikhti hai, par iska meaning alag hota hai.
Explanation
Velocity-time graph par aane wali straight slanted (teerchi) line ko samajhna kinematics ka dil hai. Jab bhi aap ek graph dekhein jisme time ke sath velocity lagaatar upar ja rahi hai, iska matlab object accelerate kar raha hai. Agar line bilkul seedhi hai (curved nahi hai), to iska matlab hai ki velocity har second mein ek barabar (equal) amount se badh rahi hai. Isey hi hum 'Uniform Acceleration' kehte hain.
Agar object exactly rest se (ruki hui halat se) shuru hota hai aur uniformly speed pakadta hai, to uski line Origin (0,0) se start hogi aur upar jayegi. Lekin agar object pehle se hi kisi speed par chal raha tha (jaise 10 m/s) aur waha se usne accelerator dabaya, to graph ki straight slanting line y-axis par 10 m/s se shuru hogi. Dono cases uniform acceleration ke hain.
Graph ka jhukav (slope) kis disha mein hai ye bhi dekhiye. Agar line upar ki jagah lagaatar straight neeche aa rahi hai (sloping downwards), iska matlab velocity waqt ke sath kam ho rahi hai, barabar hisso mein. Ise hum 'Uniform Retardation' (ya negative uniform acceleration) kahenge. Yeh tab hota hai jab driver continuously smoothly brakes apply karta hai rukne ke liye.
Kyunki velocity aur time directly proportional hote hain in uniform acceleration, ye ek straight line graph dete hain. Non-uniform acceleration ke case me (jaise unequal rate se speed badhna) line curved ho jati hai.
Worked example
Problem: A velocity-time graph for a freely falling ball shows a straight slanted line passing through the origin. What does this mean?
- Line origin se start ho rahi hai, yani initially ball ruki hui thi (u = 0).
- Line straight aur slanted hai, matlab velocity barabar rate se badh rahi hai (gravity ki wajah se).
Answer: The ball was dropped from rest and is falling with uniform acceleration.
Problem: Describe the motion shown by a v-t graph where a straight line starts at y=30 and slopes downward to meet the x-axis at x=10.
- Line y=30 se shuru hui, matlab u = 30. Line neeche x-axis par aayi, yani final velocity v = 0.
- Line strictly straight hai, jiska matlab equal retardation ho raha hai.
Answer: The object is slowing down with uniform retardation until it comes to a stop.
Key points
- Straight slanted line: A straight sloped line (inclined to the time axis) on a v-t graph indicates uniform acceleration.
- Origin vs Intercept: If it starts from origin, initial velocity is zero. If it starts from a point on the y-axis, initial velocity is not zero.
- Downward slope: A straight line sloping downwards represents uniform retardation (decreasing speed).
Common mistakes
- Confusing it with uniform motion: Student d-t graph ki aadat ki wajah se, v-t graph ki slanting line ko 'constant speed' bol aate hain. v-t graph mein slanting line ka matlab constant 'acceleration' hota hai, speed toh lagaatar badh rahi hoti hai.
- Ignoring the starting point: Har straight slanted line ka matlab rest se start hona nahi hota. Y-axis pe starting value zarur check karein ki object ruka hua tha ya pehle se chal raha tha.
Recall questions
- What does a straight slanting line on a velocity-time graph signify?
- What does a downward sloping straight line on a velocity-time graph represent?
- What is a key takeaway about interpret uniform acceleration on velocity time graph?
Questions & answers
What is the nature of the velocity-time graph for a body moving with uniform acceleration?
It is a straight line inclined to the time axis.
Approach: Since acceleration is constant, velocity increases linearly with time, forming a slanted straight line.
Draw a velocity-time graph for a stone thrown vertically upwards.
A straight line starting from a maximum positive velocity (on y-axis) and sloping downwards until it crosses the x-axis (v=0 at highest point).
Approach: The stone starts fast and slows down uniformly due to gravity, meaning it is under uniform retardation.
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