/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 A 35-kg girl is bouncing on a tr... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 36

A 35-kg girl is bouncing on a trampoline. During a certain interval after she leaves the surface of the trampoline, her kinetic energy decreases to 210 J from 440 J. How high does she rise during this interval? Neglect air resistance.

Short Answer

Expert verified
The girl rises approximately 0.67 meters.

Step by step solution

01

Understand the Problem

We need to find the height the girl rises as her kinetic energy decreases from 440 J to 210 J. This involves using the principles of energy conservation.
02

Apply Conservation of Energy

According to the conservation of energy, potential energy gained is equal to the loss in kinetic energy. Calculate the change in kinetic energy: \( \Delta KE = KE_{initial} - KE_{final} = 440 \, \text{J} - 210 \, \text{J} = 230 \, \text{J} \).
03

Relate Energy to Height

The potential energy gained can be calculated using the formula \( PE = mgh \), where \( m \) is mass, \( g \) is acceleration due to gravity (9.8 m/s虏), and \( h \) is height. Set \( PE = \Delta KE \): \( 230 \, \text{J} = 35 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times h \).
04

Solve for Height

Rearrange the equation to solve for height \( h \): \[ h = \frac{230}{35 \times 9.8} \approx 0.67 \, \text{m}. \]
05

Verify the Solution

Review the calculations to ensure the logical flow is consistent with energy conservation and check that no steps were skipped.

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Key Concepts

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

Kinetic Energy
Kinetic energy represents the energy of motion. It's the energy an object possesses due to its motion. In our trampoline scenario, the kinetic energy of the girl is the energy she has as she moves upwards thanks to her initial jump.
  • The formula for kinetic energy is: \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity.
  • Changes in kinetic energy can be observed when the speed of the moving object changes.
  • In the exercise, her initial kinetic energy was 440 J, which decreased to 210 J as she ascended.
The decrease in kinetic energy during her ascent indicates that the velocity is reducing as she climbs higher, ultimately converting kinetic energy to potential energy.
Potential Energy
Potential energy is the stored energy of an object due to its position or state. For objects raised above the ground, it relates to their vertical position and mass.
  • The gravitational potential energy formula is: \( PE = mgh \), where \( m \) is mass, \( g \) is gravity (9.8 m/s虏), and \( h \) is height.
  • In the girl's case, as her kinetic energy decreases, her potential energy increases, because she's rising higher.
  • This rise in potential energy signifies that the work done to raise her is stored as energy.
Therefore, by using the difference in kinetic energy (230 J), we can calculate how much potential energy she gained and thus how high she rose.
Energy Transformation
Energy transformation describes how energy changes from one form to another. In this exercise, the transformation is from kinetic energy to potential energy.
  • As the girl jumps upward, her kinetic energy reduces as it gets converted into potential energy.
  • The total mechanical energy (sum of kinetic and potential energy) remains the same if we neglect external forces like air resistance.
  • This principle is known as the conservation of energy, which states that the total energy in an isolated system remains constant.
By examining the changes in both forms of energy, we can predict the behavior of objects as they move and interact with forces. Hence, in the problem, we calculated the change in height by acknowledging that the loss in kinetic energy translates to a gain in potential energy.

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