Question

Spacecraft Velocity The Lunar Prospector spacecraft shown above took 105 hours to reach the Moon. Assuming that the average distance from Earth to Moon is \(384,000 \mathrm{~km}\), calculate its average velocity on the trip.

Short answer

The average velocity is approximately 3657.14 km/h.

Step-by-step solution

Identify the Given Values

We are given that the distance from Earth to the Moon is \(384,000\ \text{km}\) and the time taken by the spacecraft to reach the Moon is \(105\ \text{hours}\).

Understand the Formula for Velocity

The average velocity \(v\) is calculated by dividing the total distance \(d\) by the total time \(t\) taken, given by the formula \(v = \frac{d}{t}\).

Plug in the Values

Substitute the given values into the formula: \(v = \frac{384,000\ \text{km}}{105\ \text{hours}}\).

Perform the Calculation

Calculate the division: \(v = \frac{384,000}{105} \approx 3657.14\).

Determine the Units

The units of the velocity will be in \(\text{km/h}\) since the distance was in kilometers and the time was in hours. Therefore, the average velocity is \(3657.14\ \text{km/h}\).

Key concepts

Understanding Average Velocity

Average velocity is a key concept in physics and is crucial for calculating how fast an object is moving over a specific period of time. To determine an object's average velocity, you need to know two primary things:

• The total distance it has traveled
• The total time it took to cover that distance

The formula for average velocity is:\[v = \frac{d}{t}\]where \(v\) represents the average velocity, \(d\) stands for the distance traveled, and \(t\) is the time taken to travel that distance.
In our exercise, the Lunar Prospector spacecraft's trip provides both the distance to the Moon and the time taken, allowing us to calculate its average velocity using this straightforward formula.

The Distance to the Moon

Knowing the distance to the Moon is essential when we want to calculate the velocity of a spacecraft traveling there. The average distance from Earth to the Moon is approximately 384,000 kilometers.
This measurement is quite crucial because it represents the path our spacecraft needs to travel. Although the actual distance can vary slightly due to the Moon's elliptical orbit, for most calculations this average value is precise enough.
Considering the vastness of space, knowing such distances helps us plan and execute space missions efficiently.

Calculating Time for Space Missions

Time calculation is another critical aspect when determining the speed or velocity of an object. For space missions, knowing how long a spacecraft takes to reach its destination helps scientists make necessary adjustments and preparations.
In this exercise, the Lunar Prospector took 105 hours to reach the Moon. Understanding this time duration is essential because it allows us to use it in conjunction with the distance to calculate velocity.

The ability to calculate travel time accurately is vital for ensuring the mission's success and addressing any potential challenges during long-distance space travel.

The Art of Unit Conversion

Unit conversion is a vital skill when working with data in physics and everyday calculations. It involves changing one type of unit to another so that you can perform various mathematical operations consistently.
For our exercise, since the distance is given in kilometers (km) and time in hours (h), the resulting average velocity naturally follows as km/h or kilometers per hour.
When you work with scientific data, especially in international contexts, you must often convert units dynamically. This ensures compatibility and correctness in calculations.
Proper unit conversion prevents errors and makes sure that everyone involved in a project or study can interpret data uniformly.