What is the formula of rocket acceleration?

Rocket acceleration follows a = (F_thrust - F_drag - mg) / m - g when considering vertical launch, where F_thrust is engine thrust, F_drag is air resistance, m is current rocket mass, and g is gravitational acceleration. Unlike most acceleration problems where mass stays constant, rocket mass continuously decreases as fuel burns, making acceleration increase over time even with constant thrust. The Tsiolkovsky rocket equation governs the ultimate velocity change: Δv = v_exhaust ln(m_initial / m_final), where v_exhaust is the exhaust velocity and ln is the natural logarithm, but instantaneous acceleration requires force balance.

For a simplified analysis ignoring air resistance and gravity (deep space or very strong thrust), the formula reduces to a = F_thrust / m. If a 500,000 kg rocket produces 7,500,000 N thrust, initial acceleration is a = 7,500,000 / 500,000 = 15 m/s². After burning 200,000 kg of fuel, mass drops to 300,000 kg and acceleration increases to a = 7,500,000 / 300,000 = 25 m/s²—same thrust, higher acceleration due to lower mass. This accelerating acceleration is why rocket launches start slowly and accelerate dramatically near the end of powered flight, producing the highest g-forces just before engine cutoff.

For vertical Earth launches, the complete analysis must include gravity's opposition. The Saturn V rocket at liftoff had 33,000,000 N thrust and 3,000,000 kg mass. Net upward force was 33,000,000 - (3,000,000 × 9.8) = 33,000,000 - 29,400,000 = 3,600,000 N, giving initial acceleration a = 3,600,000 / 3,000,000 = 1.2 m/s² (or 0.12 g). This seems low, but it's deliberate too much initial acceleration stresses the structure and wastes fuel fighting gravity when the rocket is heaviest. As fuel burns and mass decreases while thrust remains relatively constant, acceleration increases to 2-3 g by upper stage separation, balancing efficiency with structural limits and crew safety. This dynamic acceleration profile distinguishes rocketry from typical constant-acceleration kinematics and requires calculus or numerical simulation for precise trajectory prediction.