Calculate mechanical energy of free falling object
Scenario
Aapne ek patthar hawa mein uchhala aur wo ek max height par jaakar thoda time rukta hai aur fir neeche girti hai.
Jab patthar upar se neeche girta hai, toh uski kinetic aur potential energy mein kya badlaav aata hai?
Mental model
Neeche girte hue, Height (`PE`) ghat-ti hai aur Speed (`KE`) badhti hai, par dono ka sum ekdum constant rehta hai.
Mechanical Energy (`PE + KE`) ek freely falling body ki poore raste constant hoti hai. Height kam hone ki wajah se `PE` ghat ti hai, par gravity use speed deti hai jisse `KE` utni hi badh jati hai. Ye perfect exchange of energy hai.
Explanation
Conservation of energy ka sabse best example ek freely falling object hai. Agar ek body mass `m` kisi height `h` par ruki hui hai, to uski velocity zero hone ke karan Kinetic Energy (`KE`) `0` hoti hai. Par Potential Energy (`PE`) full `mgh` hoti hai. Yahan total mechanical energy (`PE + KE`) hui `mgh + 0 = mgh`.
Jab object neeche girna shuru hota hai, toh uski height `h` kam hone lagti hai, jiska seedha matlab hai ki `PE` ghat rahi hai. Lekin gravity ke acceleration (`g`) ki wajah se uski velocity badhne lagti hai, yani `KE` badh rahi hai. Ek point par jitni `PE` kam hoti hai, theek utni hi `KE` badh jaati hai. Agar aap aadhi height par check karo (`h/2` par), to energy aadhi `PE` hogi aur aadhi `KE`, lekin dono ka total abhi bhi starting point wale `mgh` ke barabar hi aayega.
Zameen par takrane se theek pehle, uski height practically zero ho jati hai (so `PE = 0`). Lekin wahan velocity maximum hoti hai, toh `PE` completely `KE` mein convert ho chuki hoti hai (`1/2 m v^2 = mgh`). Isliye hum confidently keh sakte hain har ek point par `PE + KE = Constant`.
Worked example
Problem: An object of mass `20 kg` is dropped from a height of `4 m`. At a height of `2 m`, what is its potential and kinetic energy? (Assume `g = 10 m/s^2`)
- Pehle sabse top height (`4 m`) par total energy nikalte hain. Wahan `KE` `0` hoti hai aur total energy sirf `PE` hoti hai.
- Ab beech mein yani `2 m` ki height par naye wale height se `PE` nikalo.
- Kyunki total energy har jagah conserve (`800 J`) rehti hai, `KE` nikalne ke liye usme se subtract kar lo.
Answer: Potential energy is `400 J` and Kinetic energy is `400 J`.
Problem: A `5 kg` ball is dropped from a cliff. What will be its kinetic energy just before hitting the ground if its initial potential energy was `1000 J`? (Ignore air resistance).
- Law of conservation of energy kehta hai highest point ki `PE` lowest point par `KE` ban jayegi.
- Just before hitting the ground, `PE` `0` ho jayegi.
- Total energy maintain rahegi, toh ab ye saari energy `KE` ban chuki hai.
Answer: `1000 J`
Key points
- Mechanical Energy: The sum of kinetic and potential energy of an object is its total mechanical energy.
- Constant Total Energy: For a freely falling body, the total mechanical energy is constant at every point in its path.
Common mistakes
- Calculating velocity unnecessarily: Kayi bacche kisi specific height par `KE` nikalne ke liye pehle Newton ki equations (`v^2 = u^2 + 2gs`) se velocity nikalte hain fir `KE = 1/2 m v^2` karte hain. Wo lamba tarika hai! Sidha `Total Energy` mein se `PE` subtract karo, wahi bachi hui `KE` hai.
Recall questions
- What happens to the total mechanical energy of an object during free fall?
- When a ball is dropped, what is its kinetic energy exactly at the point of release?
- Just before hitting the ground, what is the potential energy of a freely falling body?
Questions & answers
A freely falling object eventually stops on reaching the ground. What happens to its kinetic energy?
It transforms into heat, sound, and potential energy of deformation.
Approach: When hitting the ground, the mechanical energy is not conserved anymore (system isn't isolated from inelastic collision), it dissipates as sound and heat.
A body of mass 10 kg is dropped from a height of 5 m. What is its kinetic energy just before reaching the ground? (g=10 m/s^2)
500 J
Approach: Loss in PE = Gain in KE. Initial PE = 10 * 10 * 5 = 500 J. So final KE = 500 J.
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