91Ó°ÊÓ

A \(1.50 \mathrm{~kg}\) water balloon is shot straight up with an initial speed of \(3.00 \mathrm{~m} / \mathrm{s}\). (a) What is the kinetic energy of the balloon just as it is launched? (b) How much work does the gravitational force do on the balloon during the balloon's full ascent? (c) What is the change in the gravitational potential energy of the balloon-Earth system during the full ascent? (d) If the gravitational potential energy is taken to be zero at the launch point, what is its value when the balloon reaches its maximum height? (e) If, instead, the gravitational potential energy is taken to be zero at the maximum height, what is its value at the launch point? (f) What is the maximum height?

Step by step solution

01

Calculate Kinetic Energy at Launch

The kinetic energy (KE) of an object is given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity. Given \( m = 1.50 \, \text{kg} \) and \( v = 3.00 \, \text{m/s} \), substituting the values gives: \[ KE = \frac{1}{2} \times 1.50 \times (3.00)^2 \] \[ KE = \frac{1}{2} \times 1.50 \times 9.00 \] \[ KE = 6.75 \, \text{Joules} \] Thus, the kinetic energy of the balloon at launch is 6.75 Joules.

02

Determine Work Done by Gravitational Force

The work done by gravity is equal to the change in gravitational potential energy, expressed as \( W = -mgh \), where \( h \) is the maximum height reached. At maximum height, all kinetic energy is converted into potential energy, hence \( mgh = KE_{initial} \). Solving \( h \), we have: \[ mgh = 6.75 \] \[ h = \frac{6.75}{1.50 \times 9.81} \approx 0.46 \, \text{m} \] Therefore, \( W = -1.50 \times 9.81 \times 0.46 \approx -6.75 \, \text{Joules} \).

03

Calculate Change in Gravitational Potential Energy

The change in gravitational potential energy (\( \Delta U \)) is equal to the initial kinetic energy, which is converted entirely into potential energy at the maximum height. Thus, \( \Delta U = 6.75 \, \text{Joules} \).

04

Gravitational Potential Energy at Maximum Height (Zero at Launch)

If the gravitational potential energy is zero at launch, then at the maximum height, the potential energy is equal to the kinetic energy at launch. Therefore, the potential energy at maximum height is 6.75 Joules.

05

Gravitational Potential Energy at Launch (Zero at Maximum Height)

If the gravitational potential energy is zero at the maximum height, the launch point must have negative potential energy matching the positive change, giving \( U_{launch} = -6.75 \, \text{Joules} \).

06

Determine Maximum Height

Using energy conservation, we found \( h = \frac{6.75}{1.50 \times 9.81} \approx 0.46 \, \text{m} \). Hence, the maximum height reached by the balloon is approximately 0.46 meters.

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Key Concepts

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

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is directly related to both the mass of the object and its velocity.
When an object is in motion, its kinetic energy can be calculated using the formula:

• \( KE = \frac{1}{2}mv^2 \)

Here, \( m \) is the mass and \( v \) is the velocity of the object. In our exercise, the water balloon has a mass of 1.50 kg and is launched with a velocity of 3.00 m/s.
By substituting these values into the formula, we found that the kinetic energy at launch is 6.75 Joules. This value tells us how much energy is contained in the moving water balloon as it begins its ascent.

Potential Energy

Potential energy is stored energy that an object has because of its position or state. For our balloon, gravitational potential energy plays a crucial role.
It depends on the height of the balloon above a reference point. Usually, the formula used is:

• \( U = mgh \)

Where \( m \) is mass, \( g \) is the acceleration due to gravity (approximated as 9.81 m/s² on Earth), and \( h \) is height above the reference point. In the exercise, the potential energy changes as the balloon rises to its maximum height.
If launch is zero, then at max height, it becomes 6.75 Joules, the entirety of the initial kinetic energy of the balloon.

Gravitational Force

Gravitational force is the attraction between two objects that have mass. For objects near Earth, such as our balloon, it pulls them towards Earth's center.
Even though gravity acts downwards, it is constantly trying to bring the balloon back to the ground. Thus, it does negative work on the balloon while the balloon goes up.

• The work done by gravity: \( W = -mgh \)

This work is equal to the change in potential energy as the balloon ascends. As seen in our specific case, the gravitational force does -6.75 Joules of work, which matches the conversion of kinetic energy into potential energy.

Work-Energy Principle

The work-energy principle is a fundamental concept in physics stating that work done on an object changes its energy.

• The principle equation: \( W = \Delta KE + \Delta U \)

In simpler terms, it suggests that energy can change forms, such as when kinetic energy turns into potential energy, but it's always conserved.
In this problem, we see this principle at work as the balloon converts all its kinetic energy (6.75 Joules) into gravitational potential energy during its climb to the peak height of approximately 0.46 meters.
Throughout its ascent, no energy is lost or gained; it's merely changing from one type to another, illustrating this principle perfectly.