Problem 53 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Take into account the Earth's rotational speed (1 rev/day) and determine the necessary speed, with respect to Earth, for a rocket to escape if fired from the Earth at the equator in a direction \((a)\) eastward; \((b)\) westward; \((c)\) vertically upward.

Step-by-Step Solution

Answer

For eastward: \(v_{east} = v_{esc} - v_e\); for westward: \(v_{west} = v_{esc} + v_e\); for upward: \(v_{vert} = v_{esc}\).

Step 1: Calculate Earth's Rotational Speed

The Earth completes one revolution per day, which is equal to 24 hours. The circumference of the Earth can be calculated as \(2\pi R\), where \(R \approx 6400 \text{ km}\). Thus, the rotational speed of the Earth is given by \(v_e = \frac{2\pi \times 6400}{24}\).

Step 2: Compute Escape Velocity

The escape velocity from Earth's surface is given by the formula \(v_{esc} = \sqrt{\frac{2GM}{R}}\), where \(G\) is the universal gravitational constant and \(M\) is the mass of the Earth. For Earth, this velocity is approximately 11.2 km/s.

Step 3: Determine Necessary Speed for Eastward Launch

When a rocket is launched eastward, it benefits from the Earth's rotational speed. Thus, the necessary speed is \(v_{east} = v_{esc} - v_e\). Compute this by subtracting Earth's rotational speed from the escape velocity.

Step 4: Determine Necessary Speed for Westward Launch

For a westward launch, the rocket is launched against Earth's rotation, so the necessary speed is \(v_{west} = v_{esc} + v_e\). Add the Earth's rotational speed to the escape velocity.

Step 5: Determine Necessary Speed for Vertical Launch

For a launch directly upward, the Earth's rotational speed does not affect the necessary escape speed. Therefore, the required speed remains \(v_{vert} = v_{esc}\).

Key Concepts

Earth's Rotational Speed

The Earth's rotational speed at the equator is crucial to understanding how it impacts rocket launches. We know that the Earth completes one full rotation every 24 hours. This rotation helps define the eastward velocity you'll experience at the equator. To figure out just how quickly the Earth is spinning, we calculate it using the formula for circumference, which is

• Formula: \[ v_e = \frac{2\pi \times 6400}{24} \]
• Where the radius \( R \approx 6400 \text{ km} \)

This equation gives us the rotational speed, roughly 0.4651 km/s. So if you're standing at the equator, you're moving eastward at this speed due to Earth's rotation.
When launching rockets, this speed is a starting boost if you go eastward, but a hurdle if you head west. For vertical launches, though, it doesn't play a major role as far as the velocity itself is concerned.

Escape Velocity Calculation

Escape velocity is the minimum speed required for an object to break free from a planet's gravitational pull without further propulsion. For Earth, we use the escape velocity formula:

• Formula: \[ v_{esc} = \sqrt{\frac{2GM}{R}} \]
• \( G \) stands for the gravitational constant, and \( M \) is Earth's mass.

Plugging in the numbers, the escape velocity comes out to be approximately 11.2 km/s for Earth.
This speed ensures that a rocket can overcome the gravitational force pulling it back. For different launch directions, this base value of 11.2 km/s adjusts according to Earth's rotational impact.

Rocket Launch Directions

When discussing rocket launches, the direction significantly affects the necessary speed to achieve escape velocity.

• Eastward Launch: Since the rocket is launched with Earth's rotation, it benefits from the Earth's rotational speed. Therefore, you reduce the escape velocity by Earth's rotational velocity: \[ v_{east} = v_{esc} - v_e \]
• Westward Launch: Here, the rocket is launched against the rotation, thus requiring more speed: \[ v_{west} = v_{esc} + v_e \]
• Vertical Launch: In this case, the Earth's rotation doesn't add or subtract from the necessary velocity functionally, maintaining the requirement of the escape velocity. The speed remains: \[ v_{vert} = v_{esc} \]

Choosing the right direction can either harness or combat Earth's rotational influence, which is critical to efficient fuel use and mission success. The added or reduced speed needs careful planning based on launch objectives and trajectory.