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Chapter 4: Problem 54

A body is falling freely under the action of gravity alone in vaccum. Which of the following quantities remain constant during the fall? (A) Kinetic energy (B) Potential energy (C) Total mechanical energy (D) Total linear momentum

Short Answer

Expert verified
The Total Mechanical Energy (C) remains constant during the free fall of a body under the action of gravity alone in vacuum, as the loss in potential energy is exactly equal to the gain in kinetic energy due to energy conservation.

Step by step solution

01

Analyze Kinetic Energy

When a body falls freely, its speed increases due to the gravitational force acting on it. The kinetic energy of an object is given by the formula: \[KE = \frac{1}{2}mv^2,\] where \(m\) is the mass of the object and \(v\) is its velocity. Since the velocity of the falling body increases during the fall, its kinetic energy does not remain constant.
02

Analyze Potential Energy

During free fall, the height of the object with respect to the Earth's surface decreases, which influences the potential energy of the body. The potential energy is given by the formula: \[PE = mgh,\] 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 level. Since the height is decreasing during the fall, the potential energy also does not remain constant.
03

Analyze Total Mechanical Energy

The total mechanical energy of the system is the sum of the kinetic energy and the potential energy: \[E_{total} = KE + PE\] As we have seen in steps 1 and 2, both the kinetic energy and the potential energy do not remain constant during the fall. However, the loss in potential energy is exactly equal to the gain in kinetic energy due to energy conservation. Therefore, the total mechanical energy remains constant during the fall.
04

Analyze Total Linear Momentum

The total linear momentum is given by the product of the mass and the velocity of the object: \[P = mv\] Since the velocity increases during free fall due to gravity, the total linear momentum does not remain constant.
05

Conclusion:

Out of the given quantities, only the Total Mechanical Energy (C) remains constant during the free fall of a body under the action of gravity alone in vacuum.

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