Analisis RK4, Improved Euler, Adams-Bashforth dan ode45 pada Fase Jatuh Bebas Bungee Jumping menggunakan MATLAB

Adhi Surya Nugraha

doi:10.31605/saintifik.v12i2.675

Adhi Surya Nugraha Program Studi Pendidikan Matematika, Universitas Sanata Dharma

DOI: https://doi.org/10.31605/saintifik.v12i2.675

Keywords: bungee jumping, metode numerik, orde konvergensi, RK4, ode45

Abstract

Fase awal bungee jumping dapat dimodelkan sebagai gerak jatuh dengan hambatan udara kuadratik. Model tersebut digunakan untuk membandingkan Runge-Kutta orde empat (RK4), Improved Euler, Adams-Bashforth orde empat, dan solver adaptif RK45 (ode45). Persamaan geraknya diturunkan dari Hukum II Newton dan memiliki solusi analitik, sehingga galat numerik dapat dihitung secara langsung. Simulasi memakai massa 68,1 kg, percepatan gravitasi 9,81 m/s², koefisien hambat 0,25 kg/m, dan interval waktu 0-10 detik. Tiga metode langkah tetap diuji pada h = 0,5, 1, dan 2 detik. Ode45 dijalankan dengan RelTol 10⁻¹⁰ dan AbsTol 10⁻¹². Pada kelompok langkah tetap, RK4 memberi galat terkecil dan orde konvergensi empiris yang mendekati empat. Adams-Bashforth masih kompetitif pada langkah kecil, tetapi galatnya bertambah lebih cepat ketika h diperbesar. Improved Euler menghasilkan galat terbesar dan menunjukkan orde empiris mendekati dua. Ode45 memberikan galat paling kecil karena langkah internalnya diatur secara adaptif. Hasil ini memperlihatkan bahwa pilihan metode perlu mempertimbangkan akurasi, ukuran langkah, dan cara solver mengendalikan galat.

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