Let a body of mass m falls from a point A, which is at a height h from the ground as shown in fig.At A,Kinetic energy kE = 0Potential energy Ep = mghTotal energy E = Ep + Ek = mgh + 0= mghDuring the fall, the body is at a position B. The body has moved a distance x from A.At B,velocity v2 = u2 + 2asapplying, v2 = 0 + 2ax = 2axKinetic energy Ek = 1/2 mv2 = 1/2 m x 2gx = mgxPotential energy Ep = mg (h – x)Total energy E = Ep + Ek = mg (h-x) + mgx = mgh – mgx + mgx= mghमाना कि M द्रव्यमान की वस्तु पृथ्वी तल से H ऊँचाई पर स्थित है l चूँकि वस्तु विराम में है So गतिज ऊर्जा Ek1 = 0 स्थितीज़ ऊर्जा = mgh स्थिति A पर का कुल ऊर्जा E1 =Ek1 +Ep1 E1=0 + mgh E1= mgh ..equation 1 अब वस्तु को गुरुत्व बल के अधीन मुक्त रूप से नीचे गिरने दिया जाता है। जब वह स्थिति A से S दूरी नीचे B पर जाती है तब पृथ्वी तल से इसकी ऊँचाई (h subtact s)Therefore Ep2 = mgh Ep2=mg(h -- S)वस्तु का पारम्भिक वेग = 0 So. V^2=u^2 +2gs(गति समीकरण)V^2=2gsB पर का कुल ऊर्जा E2 =Ek2 +Ep2 E2=1/2mv^2 + mgh E2 =mgh equation 2 अतः समीकरण 1 और 2 से तुम्हारी question prove hota h

Introduction:
When an object is freely falling, its motion is governed by the force of gravity. As the object falls, it gains kinetic energy due to its increasing speed and loses potential energy due to its decreasing height. However, the total mechanical energy of the object remains constant throughout the motion. This can be demonstrated by analyzing the potential and kinetic energy at different points during the fall.

Explanation:
1. Potential Energy: The potential energy of an object is given by the equation PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above a reference point.

2. Kinetic Energy: The kinetic energy of an object is given by the equation KE = (1/2)mv², where m is the mass of the object and v is its velocity.

3. Initial State: At the beginning of the fall, when the object is at its highest point, its velocity is zero and its potential energy is at its maximum. Therefore, the total mechanical energy is equal to the potential energy at this point.

4. During the Fall: As the object falls, its height decreases, resulting in a decrease in potential energy. However, the object gains velocity and therefore gains kinetic energy. The increase in kinetic energy exactly compensates for the decrease in potential energy, keeping the total mechanical energy constant.

5. Mid-Fall Point: At any point during the fall, the object will have a certain height, velocity, potential energy, and kinetic energy. The sum of the potential and kinetic energy at this point will be equal to the total mechanical energy, which remains constant.

6. Final State: When the object reaches the ground, its height is zero and its potential energy is also zero. However, its velocity is at its maximum, resulting in the highest kinetic energy during the fall. Again, the total mechanical energy at this point is equal to the kinetic energy.

Conclusion:
In conclusion, the sum of the potential and kinetic energy of a freely falling object remains constant at all points during its motion. This is because the decrease in potential energy is exactly offset by the increase in kinetic energy, resulting in a constant total mechanical energy. This principle is known as the conservation of mechanical energy and is a fundamental concept in physics.