Apply 2as = v^2 - u^2
Scenario
Gadi chalaate waqt road par kutta aa gaya aur driver ne full breaks maare. Skid marks (tyre ke nishaan) gadi ke rukne tak bane hue the.
Agar sirf skid marks ki lambaai (distance) aur gadi ki break ki power (retardation) pata ho, toh accident theek hone se pehle gadi ki speed kya thi, ye kaise prove karoge bina time pata kiye?
Mental model
Ye 'Timeless' equation hai. Time ka zikr kiye bina sidhe speed, acceleration aur distance ko jodti hai.
Equation 1 me displacement nahi hai, Equation 2 me final speed nahi hai. Lekin Equation 3 ek aisi math hai jo equations 1 aur 2 ko aapas me mila kar bani hai jisse `t` cut jata hai. Iska use wahan hota hai jahan time ko mapne (measure) karne wala koi na ho.
Explanation
Third equation of motion, `2as = v^2 - u^2`, kinematics ki sabse handy equation mani jati hai un situations ke liye jahan stopwatch na ho. Is equation mein total char (`4`) variable hote hain: initial velocity (`u`), final velocity (`v`), acceleration (`a`), aur displacement (`s`). Is equation ka sabse bada hallmark hai ki isme Time (`t`) ka namo-nishaan nahi hota.
Jab aapko numerical milta hai, agar question me 'how long', 'in `5 s`' jaisa koi time term na ho, aur wo initial/final conditions par baat kar raha ho (jaise bullet entering a wooden block, or car coming to a halt after skidding), toh aankh band kar ke is equation ka istemaal karein.
Mathematical errors yahan sabse zyada hote hain. Yahan `v` aur `u` dono ke squares hote hain (`v^2` aur `u^2`). Jab aapko is equation se initial velocity ya final velocity dhoondhni hoti hai, toh last step me hamesha 'square root' nikalna padta hai. Bahut sare bache square me hi answer chhod aate hain aur marks kho dete hain.
Dusri common error minus sign ki hoti hai. Jab object brakes lagata hai (comes to rest), to `v = 0` hota hai, aur equation `2as = -u^2` banti hai. Log ghabra jate hain ki distance negative kaise aayega. Par brakes ke case me acceleration (`a`) bhi negative (retardation) hota hai. Dono taraf ke negative signs last me automatically cancel ho jate hain.
Worked example
Problem: A car travelling at `20 m/s` accelerates at `2 m/s^2` over a distance of `100 m`. Find its final velocity.
- Time given nahi hai. Third equation use karenge (`2as = v^2 - u^2`).
- Multiply left side, and square `u`.
- Isolate `v^2` by adding `400` to the left side.
- Take the square root to find `v`.
Answer: `28.28 m/s`
Problem: A bullet hits a block of wood at `10 m/s` and penetrates `0.5 m` before coming to a stop. Calculate the retardation.
- Given: `u = 10 m/s`, `v = 0` (stops), `s = 0.5 m`, `t` is missing.
- Simplify both sides.
Answer: `-100 m/s^2` (meaning a retardation of `100 m/s^2`)
Key points
- The Timeless Equation: `2as = v^2 - u^2`. Use this equation exclusively when time (`t`) is not provided and not required to be calculated.
- Don't forget the square root: If you are solving for `v` or `u`, the equation gives you their squared values. You must take the square root at the final step.
- Double Negative Cancellation: When an object stops, `v = 0`, and the term becomes `-u^2`. However, since it is decelerating, `a` will also be negative, and the signs will cancel out.
Common mistakes
- Math error: (v-u)^2: Bache `2as = (v - u)^2` samajh lete hain, jo algebra me bilkul alag `(a - b)^2` identity hai. Ye exactly `v^2 - u^2` hota hai. Dono ko alag alag square karke hi subtract karna hai.
- Forgetting square root: `v^2 = 900` aa jata hai, par bache answer `v = 900` likh kar next sawal par chale jate hain. Sahi answer `v = 30` hota hai.
Recall questions
- State the third equation of motion.
- Which variable is explicitly missing from the third equation of motion?
- What is the SI unit of acceleration?
Questions & answers
A train is travelling at a speed of 90 km h-1. Brakes are applied so as to produce a uniform acceleration of -0.5 m s-2. Find how far the train will go before it is brought to rest.
625 m
Approach: u = 90 km/h = 25 m/s. v = 0. a = -0.5. Since t is not needed, use 2as = v^2 - u^2. 2 * (-0.5) * s = 0^2 - 25^2. -1 * s = -625. s = 625 m.
A scooter moving at 10 m/s is stopped by applying brakes which produce a uniform retardation of 0.5 m/s^2. How much distance will be covered by the scooter before it stops?
100 m
Approach: u = 10, v = 0, a = -0.5. 2as = v^2 - u^2 -> 2 * (-0.5) * s = 0 - 100 -> -s = -100 -> s = 100 m.
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