Vertical Motion: Distance Decreases Arithmetically Until Maximum Height

What This Problem Teaches

• Recognizing arithmetic sequences in physical motion contexts
• Finding when a decreasing sequence reaches zero
• Calculating sums of arithmetic sequences using formulas
• Converting between discrete intervals and continuous time
• Understanding the relationship between distance traveled and maximum height in projectile motion

Visualizing the Motion

Solution: Method 1 — Arithmetic Sequence Analysis

Step 1 — Identify the pattern

The distances form an arithmetic sequence: 33, 30, 27, 24, ... where each term decreases by 3 inches. This is an arithmetic sequence with first term a₁ = 33 and common difference d = -3.

Step 2 — Find the general term formula

For an arithmetic sequence, the nth term is aₙ = a₁ + (n-1)d. Substituting our values:

Step 3 — Find when the distance becomes zero

The ball starts coming down when the distance traveled in an interval becomes zero or negative. We need the last positive term, so we set aₙ = 0:

So the 12th interval is when the distance becomes zero. This means maximum height is reached at 12 × 0.1 = 1.2 seconds.

Step 4 — Calculate total distance to maximum height

We need to sum the first 11 positive terms (intervals 1 through 11), since the 12th interval has zero distance. Using the arithmetic series formula S = n(a₁ + aₙ)/2:

First, find the 11th term:

Now calculate the sum:

Solution: Method 2 — Direct Summation Approach

Step 1 — List all positive distances

Instead of using the formula, let's directly calculate each term until we reach zero:

Step 2 — Add the positive distances

Sum intervals 1 through 11:

Step 3 — Confirm timing

The ball reaches zero upward velocity at the end of interval 12, which occurs at 12 × 0.1 = 1.2 seconds.

Verification

Check the arithmetic sequence formula

Let's verify our 12th term calculation:

Check the sum using the alternative arithmetic series formula

We can also use S = n(2a₁ + (n-1)d)/2 for the first 11 terms:

Physical reality check

The ball travels about 16.5 feet upward in 1.2 seconds, which is reasonable for a ball thrown with significant initial velocity. The decreasing distances make sense as gravity slows the ball's upward motion.

Common Pitfalls

The Pattern Behind This

This problem illustrates a key principle in physics disguised as pure arithmetic. When an object moves under constant acceleration (like gravity), the distances traveled in equal time intervals form an arithmetic sequence.

In our case: d₀ = 33, k = 3, so n = 33/3 + 1 = 12 intervals, reaching maximum at 12 × 0.1 = 1.2 seconds.

Where This Shows Up in Real Life

• Sports analysis: Basketball shot trajectories are analyzed using similar discrete time intervals to optimize shooting angles and understand ball flight patterns.
• Engineering: Rocket trajectory calculations use step-wise approximations when continuous integration is computationally expensive.
• Video game physics: Game engines often use discrete time steps (like 1/60th of a second) to calculate projectile motion, creating arithmetic sequences in velocity and position updates.

What If?