A ball is dropped from a height of 100 meters. Each time it bounces, it reaches 60% of its previous height. Calculate the total vertical distance traveled by the ball after it hits the ground for the fifth time.

["Title: How High Does a Ball Bounce After Dropping from 100 Meters? Calculating Total Vertical Distance After Five Bounces", "When a ball is dropped from a height of 100 meters, the motion involves repeated bouncing, with each bounce reaching 60% of the previous height. Understanding the total vertical distance traveled is essential in physics, engineering, and sports science. This article calculates the total vertical distance the ball travels after it hits the ground for the fifth time, including both upward and downward travel, using principles of geometric sequences.", "---", "### What Happens When a Ball Is Dropped and Bounced?", "Each bounce follows a predictable pattern: the ball falls vertically, hits the ground, and springs back to 60% of its previous height. After each bounce, two distances are added to the total:\n- The downward distance (during the drop and each bounce down)\n- The upward distance (after each bounce, before the next drop)", "However, after each bounce except the first drop, the ball travels both up and down. Since we are asked for the total distance traveled after the fifth ground hit, we include all motions up to and including the descent on the fifth impact.", "---", "### Step 1: Initial Drop", "- Height dropped from: 100 m (only downward)\n- Distance so far: 100 m\n- Bounce count: 0 (first drop — not a bounce)", "---", "### Step 2: First Bounce (After 1st hit)", "- Bounce height = 60% of 100 m = 60 m\n- Upward distance on first bounce: 60 m\n- Downward distance: 60 m (next impact)\n- Cumulative vertical distance:\n (100 + 60 + 60 = 220) m\nBut we only want distances until the fifth ground hit, so keep track per bounce.", "We compute total distance after each impact (lands on ground), counting both upward and downward legs until the fifth ground contact.", "---", "### Modeling the Bounce Sequence", "Let ( h_0 = 100 ) m (initial height)", "At each bounce ( n \geq 1 ), the rebound height is:\n[ h_n = 0.6 \cdot h_{n-1} ]", "But the ball travels up and down each bounce except after the initial drop. However, before the first bounce (after drop), it falls 100 m. After first bounce, it rises 60 m and falls 60 m, and so on.", "So the total distance after ( n ) ground hits is:", "[\nD_n = h_0 + 2 \sum_{k=1}^{n} h_k\n]", "Because:\n- First segment: constant 100 m drop → 100 m\n- Then, each bounce contributes ( 2 \ imes h_k ) (up and down), except we don’t double count the final drop in this case — but since we stop at fifth ground hit, the last bounce includes an upward path but no final downward drop beyond that.", "Thus, after five hits:\n- First drop: 100 m\n- Then: drop + bounce, up + down, repeating for 4 bounces (giving 5 hits total)", "So total distance:", "[\nD = 100 + 2(h_1 + h_2 + h_3 + h_4)\n]", "Now compute each:", "- ( h_1 = 100 \ imes 0.6 = 60 )\n- ( h_2 = 60 \ imes 0.6 = 36 )\n- ( h_3 = 36 \ imes 0.6 = 21.6 )\n- ( h_4 = 21.6 \ imes 0.6 = 12.96 )", "Sum of rebound heights (up trips):\n[\n60 + 36 + 21.6 + 12.96 = 130.56\n]", "Multiply by 2 for up and down:\n[\n2 \ imes 130.56 = 261.12\n]", "Add initial drop:\n[\nD = 100 + 261.12 = 361.12 \ ext{ meters}\n]", "---", "### Verification of Bounce Sequence", "Let’s trace the drops and bounces to confirm:", "1. Drop 1: Fall 100 m (ground hit #1)\n2. Rise: 60% of 100 = 60 m (accounted — new peak)\n3. Fall: 60 m (ground hit #2)\n4. Rise: 60 × 0.6 = 36 m\n5. Fall: 36 m (ground hit #3)\n6. Rise: 36 × 0.6 = 21.6 m\n7. Fall: 21.6 m (ground hit #4)\n8. Rise: 21.6 × 0.6 = 12.96 m\n9. Fall: 12.96 m → ground hit #5", "Total vertical distance:\n[\n100 + 60 + 60 + 36 + 36 + 21.6 + 21.6 + 12.96 + 12.96 = ?\n]", "Group:\n- Initial: 100\n- Repeated distances:\n (60×2) + (36×2) + (21.6×2) + (12.96×2) =\n (2(60 + 36 + 21.6 + 12.96) = 2(130.56) = 261.12)", "Total:\n[\n100 + 261.12 = 361.12 \ ext{ meters}\n]", "---", "### Important Note: Why Not Include Final Drop?", "Although the ball reaches a rebound height of (12.96 \ imes 0.6 = 7.776) m on the fifth bounce, this upward motion occurs before the fifth ground hit. Since the fifth hit occurs on the downward motion of the fourth bounce (to 12.96 m), we include that full 12.96 m descent. The rebound after that would be the sixth hit — which we do not compute.", "---", "### Summary", "After a ball is dropped from 100 meters:", "- It falls 100 m (1st impact)\n- Rises to 60 m, falls 60 m (2nd impact)\n- Rises to 36 m, falls 36 m (3rd impact)\n- Rises to 21.6 m, falls 21.6 m (4th impact)\n- Rises to 12.96 m, falls 12.96 m (5th impact)", "Total vertical distance traveled after the fifth ground hit is:", "[\n\boxed{361.12} \ ext{ meters}\n]", "---", "### Applications", "This model helps engineers design safety equipment, optimize jump performance in sports, and simulate bouncing dynamics. Using geometric series, we efficiently calculate motion without tracking each frame — a powerful tool in physics simulation and real-world applications.", "---", "Keywords:\nBall dropped from 100 meters, bounce height 60%, total vertical distance, after fifth ground hit, exact distance calculation, physics bounce problem, geometric series, vertical motion, projectile rebound, distance after each impact"]