Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Assuming air resistance is negligible, a small object that is dropped from a hot air balloon falls 16 feet during the first second, 48 feet during the second second, 80 feet during the third second, 112 feet during the fourth second, and so on. Find an expression for the distance the object falls in \(n\) seconds.

Step-by-Step Solution

Answer

The distance fallen in n seconds is given by \(d = 16n^2\).

Step 1: Identify the Pattern

Observe the given distances: 16, 48, 80, and 112 feet for the first four seconds. Notice the pattern that the difference between each successive distance increases by a constant value of 32 feet.

Step 2: Express the Sequence Formula

Based on the pattern identified, the nth term of the sequence can be expressed as a linear arithmetic series. Notice that each term increases by 32, indicating a common difference.

Step 3: Define the Arithmetic Sequence

The addition created after the first term (16 in this case) can be identified with an arithmetic sequence formula: \( a_n = a_1 + (n-1)d \), where \(d = 32\) and \(a_1 = 16\).

Step 4: Derive General Formula for Distance in n Seconds

Using the formula \( a_n = 16 + (n-1) \times 32 \), we can sum the first n terms to find the total distance fallen after n seconds. This is because each term is the distance fallen in a specific second.

Step 5: Apply the Sum of Arithmetic Sequence Formula

The distance fallen in n seconds is the sum of the first n terms of the arithmetic sequence. Use the sum formula for an arithmetic series: \( S_n = \frac{n}{2} (a_1 + a_n) \).

Step 6: Calculate the Total Distance Formula

Plugging into the sum formula, we find \( S_n = \frac{n}{2} [16 + (16 + (n-1)\times32)] \), simplifying to \( S_n = \frac{n}{2} (16 + 16 + 32(n-1)) = \frac{n}{2} (32n) \).

Step 7: Simplify the Expression

The formula simplifies further to \( S_n = 16n^2 \), which represents the total distance fallen in n seconds.

Key Concepts

Arithmetic series

An arithmetic series is the total of the terms in an arithmetic sequence. In our scenario, where a falling object has increasing distances with each second, this is a perfect concept to apply. The object's distance increases regularly by 32 feet each second, creating an arithmetic sequence.

• An arithmetic sequence is a set of numbers with a constant difference between consecutive terms.
• An arithmetic series adds all terms in this sequence.

In our exercise, the sequence is 16, 48, 80, and so on, with 32 being the constant difference. To determine the total distance, we must sum these individual distances, which involves using the sum of the sequence.

Sum of arithmetic sequence

The sum of an arithmetic sequence calculates the total by adding all terms of the sequence. For our falling object, we use it to calculate the total distance fallen after several seconds without air resistance. The formula to compute the sum of an arithmetic sequence is:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Where:

• \( n \) is the number of terms.
• \( a_1 \) is the first term of the sequence.
• \( a_n \) is the nth term.

After recognizing that the difference between terms is 32 feet and using the formula for the nth term \( a_n = a_1 + (n-1)d \), we can derive the overall sum, simplifying finally to \( S_n = 16n^2 \). This gives us the total distance.

Distance and velocity without air resistance

In physics, understanding distance and velocity without air resistance allows us to calculate how an object falls purely under gravity's influence.

• The distance an object falls is a direct result of acceleration due to gravity.
• In this exercise, distances are calculated over successive seconds as an arithmetic sequence.

Without air resistance, the calculation becomes simpler as it eliminates any drag forces. So, the focus is solely on gravity, which translates into the constant addition in distance each second. In this case, it becomes 32 feet more each second after the initial 16-foot drop.

Physics applications in mathematics

Math is vital in solving real-world physics problems, like calculating distances over time without air resistance. Our exercise showcases such an application, where arithmetic sequences model a falling object's path.
Here's how math and physics intersect:

• We can predict motion and calculate distance using arithmetic sequences and series.
• The consistent acceleration due to gravity forms a basis for such calculations.

By simplifying the problem to math's fundamental operations, we can derive useful formulas, like \( S_n = 16n^2 \), as shown. This highlights how mathematics helps us understand continuous motion in physics, making it easier to predict outcomes of similar problems.