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Chapter 6: Problem 77

A 2.00-kg rock is released from rest at a height of 20.0 m. Ignore air resistance and determine the kinetic energy, gravitational potential energy, and total mechanical energy at each of the following heights: 20.0, 10.0, and 0 m.

Short Answer

Expert verified
At each height, total mechanical energy is 392.2 J.

Step by step solution

01

Define Gravitational Potential Energy

The gravitational potential energy (GPE) at a height is given by the formula \( U = mgh \), where \( m \) is the mass (2.00 kg), \( g \) is the acceleration due to gravity (9.81 m/s虏), and \( h \) is the height. Calculate \( U \) for each height: 20.0 m, 10.0 m, and 0 m.
02

Compute Gravitational Potential Energy at 20.0 m

Using the formula for GPE, \( U = (2.00\, \text{kg})(9.81\, \text{m/s}^2)(20.0\, \text{m}) \), we find \( U = 392.2 \) J.
03

Compute Gravitational Potential Energy at 10.0 m

At 10.0 m, using the same formula, \( U = (2.00\, \text{kg})(9.81\, \text{m/s}^2)(10.0\, \text{m}) \), we find \( U = 196.2 \) J.
04

Compute Gravitational Potential Energy at 0 m

At 0 m, the GPE is \( U = (2.00\, \text{kg})(9.81\, \text{m/s}^2)(0\, \text{m}) = 0 \) J.
05

Define Kinetic Energy

The kinetic energy (KE) is given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity. At each height, the velocity can be determined using energy conservation between initial potential energy and current height.
06

Calculate Total Mechanical Energy

The total mechanical energy (TME) is constant and given by the initial total energy, which is all potential energy at the start: \( TME = 392.2 \) J.
07

Determine Kinetic Energy at Each Height

KE can be determined from \( TME = KE + U \). At 20.0 m, KE = 0 J. At 10.0 m, \( KE = 392.2 - 196.2 = 196.0 \) J. At 0 m, \( KE = 392.2 - 0 = 392.2 \) J.
08

Summarize Results

At 20.0 m: \( KE = 0 \) J, \( U = 392.2 \) J, \( TME = 392.2 \) J. At 10.0 m: \( KE = 196.0 \) J, \( U = 196.2 \) J, \( TME = 392.2 \) J. At 0 m: \( KE = 392.2 \) J, \( U = 0 \) J, \( TME = 392.2 \) J.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Potential Energy
Gravitational Potential Energy (GPE) is a type of energy possessed by an object due to its position relative to the Earth. It can be thought of as energy stored because of an object鈥檚 height. The formula to calculate GPE is \( U = mgh \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximately 9.81 m/s虏 on Earth), and \( h \) is the height above the ground.
To solve for GPE at different heights:
  • At 20.0 m, GPE is calculated as \( 392.2 \) J.
  • At 10.0 m, GPE decreases to \( 196.2 \) J.
  • Finally, at 0 m, the GPE becomes 0 J.
This shows that as the rock falls, its gravitational potential energy decreases due to the reduction in height.
Kinetic Energy
Kinetic Energy (KE) is the energy an object possesses because of its motion. The formula for calculating kinetic energy is \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object. As an object moves, this energy is in use.
In the scenario of the falling rock:
  • At 20.0 m, the rock hasn鈥檛 started moving yet; hence, its kinetic energy is 0 J.
  • At 10.0 m, it has started to fall, converting some potential energy to kinetic energy, with \( KE = 196.0 \) J.
  • At 0 m, the rock's kinetic energy reaches \( 392.2 \) J as all potential energy is converted to kinetic energy.
This transformation of energy from potential to kinetic, as the rock gains speed, illustrates the dynamic nature of energy in motion.
Energy Conservation
Energy Conservation is a fundamental principle stating that the total energy of an isolated system remains constant. In simpler terms, energy cannot be created or destroyed; it can only be transformed or transferred from one form to another.
For the falling rock, the total mechanical energy (TME) is conserved throughout the fall. Initially, all of its energy is gravitational potential energy because it starts from rest at the top. As it falls:
  • The total mechanical energy at 20.0 m is \( 392.2 \) J, purely potential energy.
  • At 10.0 m, the total mechanical energy remains \( 392.2 \) J, split between kinetic (196.0 J) and potential (196.2 J).
  • At 0 m, although all potential energy is converted to kinetic energy, the total remains \( 392.2 \) J.
This conservation shows that mechanical energy provides consistency in the intricacies of motion and forces. It confirms that as one form of energy decreases, the other increases, keeping the total energy constant.

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