Problem 2 - Calculus Single Variable

Question

On the surface of the earth, a rocket weighs 10,000 newtons. How much work is performed lifting the rocket to a height 100 kilometers above the surface of the earth, assuming that the direction of the force, as well as the motion, is straight up?

Step-by-Step Solution

Answer

The work done is 1,000,000,000 joules.

Step 1: Understand Work Formula

Work is calculated using the formula: \( W = F \cdot d \cdot \cos(\theta) \), where \( W \) is the work done in joules, \( F \) is the force in newtons, \( d \) is the distance over which the force is applied in meters, and \( \theta \) is the angle between the force direction and the direction of motion. In this problem, \( \theta = 0 \) degrees since the force and motion are both straight up, making \( \cos(\theta) = 1 \).

Step 2: Convert Distance to Meters

The problem gives the height the rocket is lifted as 100 kilometers. To use this in the work formula, convert kilometers to meters: 100 kilometers is 100,000 meters.

Step 3: Apply Values to the Work Formula

Substitute the given values into the work formula: \( W = 10,000 \, \text{N} \times 100,000 \, \text{m} \times 1 = 1,000,000,000 \, \text{J} \). This means the work done is 1 billion joules.

Key Concepts

Force and Its Role in Work

When discussing work and energy, force is a fundamental concept. In the context of this exercise, force refers to any interaction that, when unopposed, will change the motion of an object. In simple terms, force is what pushes or pulls an object.

In physics, especially when calculating work, force is measured in newtons. For our rocket example, a force of 10,000 newtons is used to lift the rocket. This force is consistent and opposes the downward pull of gravity.

Understanding that force is consistently applied is key. It directly affects how much work is done:

• The more force you apply, the more work you do if the object moves over the same distance.
• If the force doesn’t cause the object to move, no work is done regardless of the magnitude of the force.

In the work formula, force multiplies with distance to compute total work, crucial for determining the energy expended in lifting the rocket.

Importance of Distance in Work Calculation

Distance is another vital part of understanding work in physics. It measures how far the force has moved the object. In our formula for work, the distance is the path over which the object is moved by the force.

For the rocket problem, we need to consider the distance from the surface of the Earth to 100 kilometers above. Originally presented in kilometers, this is crucial because physics calculations often use the metric system, which requires converting this distance into meters. Since 1 kilometer equals 1,000 meters, 100 kilometers equals 100,000 meters.

Why is distance so important?

• Longer distances mean more potential work done, assuming constant force.
• Distances need to be consistent with the unit system used for other variables, like force in newtons.

Overall, understanding how to accurately measure and convert distances allows you to precisely calculate the work done.

Connecting Physics Concepts

Physics encompasses many interrelated concepts like energy, force, and distance, which cumulatively explain natural phenomena. When calculating work, we rely heavily on these principles, integrating knowledge from multiple areas of physics.

Understanding work in the context of our exercise involves understanding:

• Energy, as work is a form of energy transfer.
• Forces, which initiate the movement of the rocket.
• Distance, over which this force is applied aligns with our path of movement.

Together, these aspects lay the foundation of classical mechanics. Namely, they allow us to predict the movement, energy use, and interactions of objects under various forces.

The underlying physics of lifting the rocket encompasses not just the immediate force applied but also how that force converts into work and subsequently into the potential energy associated with height—it is all a dance of energy transformation prescribed by physics laws.