Problem 77 - A Graphical Approach to College Algebra

Question

Solve each problem. The velocity \(v\) of a meteorite approaching Earth is given by $$v=\frac{k}{\sqrt{d}}$$ measured in kilometers per second, where \(d\) is its distance from the center of Earth and \(k\) is a constant. If \(k=350\) what is the velocity of a meteorite that is 6000 kilometers away from the center of Earth? Round to the nearest tenth.

Step-by-Step Solution

Answer

The velocity is approximately 4.5 km/s.

Step 1: Understand the Formula

The velocity of the meteorite is given by the formula \[v = \frac{k}{\sqrt{d}}\]where \(v\) is the velocity, \(k\) is a constant, and \(d\) is the distance from the center of the Earth.

Step 2: Identify Given Values

From the problem, we know that \(k = 350\) and \(d = 6000\) kilometers.

Step 3: Substitute Values into the Formula

Substitute \(k = 350\) and \(d = 6000\) into the formula to find the velocity:\[v = \frac{350}{\sqrt{6000}}\]

Step 4: Calculate the Square Root

Evaluate \(\sqrt{6000}\):\[\sqrt{6000} \approx 77.46\]

Step 5: Compute the Velocity

Now divide 350 by the square root of 6000:\[v = \frac{350}{77.46} \approx 4.52\]

Step 6: Round to the Nearest Tenth

Round the calculated velocity to the nearest tenth, which gives:\[v \approx 4.5\]

Key Concepts

Understanding Meteorite Velocity

The velocity of a meteorite, when it approaches Earth, helps us understand how fast it is moving. This speed is significant because it tells us about the energy and force with which a meteorite can impact Earth.
The simple formula given is:

• \(v = \frac{k}{\sqrt{d}}\)

Here, \(v\) is the velocity, \(k\) is a constant value specific to the conditions of the meteorite and its path, and \(d\) represents the distance from the Earth's center.
By inserting the value of \(d\), you can determine how fast the meteorite is moving as it gets closer. This expression simplifies what otherwise could seem like a daunting physics problem.

Distance From Earth's Center

Distance plays a crucial role in the velocity of an approaching meteorite. This distance is measured from the Earth's center, not from its surface, which might initially seem confusing.
Earth's radius is approximately 6371 kilometers, so any distance \(d\) from the center would be greater than this value. However, for our practical calculation:

• This distance is given simply as 6000 kilometers, placing it fairly close in the context from which the Earth’s atmosphere begins to thin.
• The closer \(d\) is to the center, the higher the velocity \(v\) becomes, as seen from the formula.

As \(d\) decreases, \(\sqrt{d}\) reduces, which increases the overall speed, emphasizing the inverse square relationship between distance and velocity.

Executing a Square Root Calculation

Square root calculations are pivotal to many physics problems, including our meteorite velocity equation. Here’s how we identify and compute them:

• First, understand that \(\sqrt{6000}\) is asking "What number, multiplied by itself, equals 6000?"
• Using a calculator, you find \(\sqrt{6000} \approx 77.46\).
• This component of the formula must be calculated accurately because any error affects the result significantly.

With this precision, meteorite velocity is calculated correctly, allowing us to make informed predictions and take necessary precautions.