Problem 76 - Precalculus: Mathematics for Calculus

Question

When an object is allowed to fall freely near the surface of the earth, the gravitational pull is such that the object falls \(16 \mathrm{ft}\) in the first second, \(48 \mathrm{ft}\) in the next second, \(80 \mathrm{ft}\) in the next second, and so on. (a) Find the total distance a ball falls in 6 s. (b) Find a formula for the total distance a ball falls in \(n\) seconds.

Step-by-Step Solution

Answer

(a) 576 ft; (b) Total distance in n seconds: \( S_n = 16n(n + 1) \).

Step 1: Understanding the Problem

We need to find the total distance fallen by an object in a given time when it is under free fall with increasing distances covered every second. The distance covered by the object in each second forms an arithmetic sequence.

Step 2: Identifying the Sequence for One Second Intervals

The distances fallen each second can be noted as 16 ft in the first second, 48 ft in the second second, 80 ft in the third second, and so on. The sequence of distances is 16, 48, 80,... This sequence has a common difference, and we need to identify this.

Step 3: Determining the Common Difference

To find the common difference, calculate the difference between successive terms, such as 48 - 16 = 32 and 80 - 48 = 32. The common difference is 32, confirming that this is an arithmetic sequence.

Step 4: Calculating Total Distance for 6 Seconds

The total distance fallen is the sum of distances each second, which is the sum of the first six terms of the arithmetic sequence. We use the formula for the sum of the first n terms of an arithmetic sequence: \( S_n = \frac{n}{2} (2a + (n-1)d) \), where \( a = 16 \) (first term) and \( d = 32 \) (common difference).

Step 5: Applying the Formula for 6 Seconds

Substitute \( n = 6 \), \( a = 16 \), and \( d = 32 \) into the formula: \( S_6 = \frac{6}{2} (2 \cdot 16 + (6-1) \cdot 32) \). Calculate \( S_6 = 3 (32 + 160) = 3 \times 192 = 576 \). The total distance fallen in 6 seconds is 576 ft.

Step 6: Deriving General Formula for n Seconds

To find a general expression for any number of seconds \( n \), consider the sum formula: \( S_n = \frac{n}{2} (2 \cdot 16 + (n-1) \cdot 32) \). Simplifying: \( S_n = \frac{n}{2} (32 + 32n - 32) = 16n(n + 1) \).

Step 7: Verifying General Formula

The formula \( S_n = 16n(n + 1) \) should yield consistent results. For \( n = 6 \), plug into the formula, getting \( 16 \times 6 \times 7 = 576 \), confirming that it matches the earlier calculated value.

Key Concepts

Free Fall

Free fall describes the movement of an object falling solely under the influence of gravity. It's a fascinating concept in physics! When an object is in free fall near the Earth's surface, it accelerates downwards due to Earth's gravitational pull. The object doesn't encounter any resistance like air drag, assuming an ideal scenario.

Here's what's intriguing about free fall:

• The acceleration remains constant at approximately 32 ft/s² on Earth.
• This acceleration is due to the gravitational pull from Earth, impacting all objects equally regardless of their mass.
• In our exercise, the object's falling distances per second depict an increasing arithmetic pattern, a direct result of this constant acceleration.

Understanding free fall helps explain why objects speed up as they fall and can be calculated accurately using arithmetic sequences to find the distances fallen each second.

Gravitational Pull

Gravitational pull is a natural phenomenon by which every object with mass attracts another with a force proportional to their masses and inversely proportional to the square of the distance between them. For objects near Earth's surface, this force results in a downward acceleration known as gravity.

For practical purposes, gravitational pull can:

• Accelerate objects downward at a steady rate, which, on Earth, is about 32 ft/s².
• Influence the fall of objects in an arithmetic sequence pattern due to this constant acceleration force.
• Enable us to calculate the distances covered in free fall using mathematical formulas.

In simple terms, gravitational pull is why we stay grounded and why every object falls towards the Earth when dropped. In exercises like the one given, understanding gravitational pull means interpreting how kids from different distances form an arithmetic sequence.

Arithmetic Sequence Formula

An arithmetic sequence is a series of numbers in which the difference between consecutive terms remains constant. In our exercise scenario, the sequence—16, 48, 80, and so on—has a consistent difference, revealing an arithmetic sequence pattern.

Here's how you understand and use arithmetic sequences:

• The sequence starts with a beginning term, often denoted by **a**.
• Each following term increases by a constant difference called **d**.
• The general formula for any term in an arithmetic sequence is given by \( a_n = a + (n-1) \times d \).
• To find the sum of the first **n** terms in the sequence, you use a specific formula: \( S_n = \frac{n}{2} (2a + (n-1)d) \).

Recognizing and applying these formulas allows us to determine distances like those in the free fall problem, where each second’s fall distance forms a predictable pattern through arithmetic sequencing.