Rocket launch - 2D trajectory and delta-v budget

The rocket equation

A rocket gains speed by throwing mass backwards. With exhaust speed v_e = I_sp g₀, momentum conservation gives the Tsiolkovsky equation Δv = v_e ln(m₀ / m_f). The propellant mass flow is ṁ = F / (I_sp g₀), and lift-off needs F > m g, i.e. a thrust-to-weight ratio above 1.

Losses and the gravity turn

The speed actually reached is the ideal Δv minus the gravity loss ∫ g sinγ dt (the engine spends effort just holding the rocket up) and the drag loss ∫ D/m dt, with D = ½ ρ v² C_D A and ρ = 1.225 e^{-h/8.5 km}. Flying vertically wastes the most on gravity, so after a short pitch kick the thrust follows the velocity vector (a gravity turn) and the path bends over without side forces. Dynamic pressure q = ½ ρ v² peaks at max-Q, where the structure is loaded most.

Orbit needs about 7.8 km/s sideways at 200 km, and total ideal Δv of roughly 9 to 9.5 km/s once the losses are paid, which is why real launchers use several stages.

Honest simplifications: one stage of fixed mass 100 t, constant thrust with a mild pressure-dependent I_sp (up to 10 % lower at sea level), an isothermal exponential atmosphere, a constant drag coefficient, a flat downrange coordinate corrected only by the centripetal term v_x²/(R+h), no wind, no staging and an instantaneous pitch kick. It is a teaching model of launch physics, not a trajectory design tool.

• Set the pitch kick to 0 and compare the gravity loss with a 2 degree kick: bending the path early cuts the loss.
• Lower the thrust-to-weight to 1.1: the burn is long and the gravity loss large.
• Raise the specific impulse to 450 s (hydrogen) and watch the burnout speed approach the orbital 7.8 km/s line.
• Compare the peak dynamic pressure of a sounding rocket (TWR 4) with the launcher.