A rocket throws hot exhaust gas backwards very fast; the gas pushes back on the rocket with an equal and opposite force, which drives it forwards.
Newton's third law says that when one object pushes on a second object, the second pushes back equally hard in the opposite direction. Every action has an equal and opposite reaction.
A rocket engine is built around this law. It burns propellant in a chamber and forces the hot gas out through a nozzle at the back. The engine pushes the gas backwards. The gas pushes the engine forwards. That forward push is called thrust.
A common mistake is to think a rocket pushes against the ground or the air behind it. It does not. The push comes from the exhaust it throws out. Because a rocket carries its own oxidiser, it can do this in a vacuum, where there is no air at all.
In fact, outside air gets in the way slightly. The thrust equation has a term that depends on the difference between the exhaust pressure and the outside pressure. In vacuum the outside pressure is zero, so the same engine pushes a little harder. ISRO's SSLV figures, for example, are given as maximum vacuum thrust.
LVM3 has two S200 strap-on boosters. Each holds about 205 tonnes of solid propellant and burns for about 134 seconds. On average, each one throws out roughly 1.5 tonnes of exhaust every second.
At lift-off these two boosters are the only engines firing. The liquid core stage, with its two Vikas engines, lights only about 113 seconds into the flight, at an altitude of about 41.5 km. So the reaction to the S200s' exhaust is what first lifts the whole 640-tonne rocket off the pad.
The law works the same way for big and small engines. On PSLV, the Vikas engine of the second stage gives up to 799 kN. The two small fourth-stage engines give about 7.3 kN each. SSLV's final module uses thrusters of just 50 N to fine-tune its speed.
Even a satellite in orbit moves the same way: its thrusters throw out gas and the satellite is pushed the other way.
Picture the two S200 boosters on each side of the core — at lift-off they supply all of LVM3's thrust. See LVM3's flights
Thrust comes from momentum. In a short time dt the engine throws out a mass dm of gas at speed Ve relative to the rocket. The gas gains momentum Ve·dm backwards, so the rocket gains the same momentum forwards: M·dv = Ve·dm. Dividing by dt gives the force: F = ṁ·Ve, where ṁ is the mass flow rate.
NASA's full rocket thrust equation adds a pressure term:
F = ṁ·Ve + Ae·(pe − p0)
Here Ae is the nozzle exit area, pe the exhaust pressure at the exit and p0 the outside pressure. As the rocket climbs, p0 falls towards zero and the pressure term grows. The same engine gives more thrust in vacuum than at sea level.
Worked numbers for LVM3:
This is only an average estimate from ISRO's published figures, but it shows the pattern. The boosters move a lot of mass at a lower speed. The cryogenic upper stage moves little mass at a high speed.
Integrate M·dv = Ve·dm from full mass to empty mass and you get the rocket equation, Δv = Ve·ln(m_full / m_empty). The third law is where that equation starts.
When one object exerts a force on another, the second exerts an equal and opposite force on the first. The engine pushes exhaust gas backwards (action), and the gas pushes the rocket forwards (reaction).
The thrust equation contains the term Ae(pe − p0). In vacuum the outside pressure p0 is zero, so this term, and the total thrust, is larger.
Only the two S200 solid boosters. The L110 liquid core stage ignites about 113 seconds later, at about 41.5 km altitude.