GRADES 9–12
NGSS: Motion and Stability:
Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration.
If you have read through the above sections, you probably have a decent idea of the fundamentals of rocket propulsion. With this understanding, we can derive one of the most fundamental relations in rocketry – the thrust equation. We begin by drawing a “free body diagram” (a
picture showing the forces and where they are applied) of a rocket in flight.
We want to find an expression for the thrust which is driving the rocket forwards; we know that along the primary axis of the rocket, we have the force of the rocket’s exhaust gases pushing on the rocket; we also know that as the exhaust is forced out of
the engines, momentum is leaving the rocket (this is not itself a force, but is shown on the free body diagram for completeness).
Newton’s Second Law tells us that the net force on an object is equal to the product of its mass and its acceleration. With a change in the object’s mass, we consider the rate at which momentum is leaving the object to be a fictitious “force.” (Formally speaking, we could
invoke Euler’s First Law, which generalizes Newton’s Second Law to include variation in mass and which states that the net force on an object is equal to the rate of change in its momentum.) We express Newton’s Second Law as:
We can write each of these terms as follows:
where A_{e} is the exit area of the rocket’s engines (the area over which the pressure force is acting), P_{e} is the exit pressure of the exhaust gases, and P_{a}is the ambient atmospheric pressure (zero when the rocket is in space). The rocket’s exhaust exerts a pressure on the
rocket, but the ambient atmosphere does as well, which is why we need to take the difference of the two pressures. The momentum term is written as:
where ṁ is the mass flow rate (amount of mass per unit time) exiting the engines, V_{gas} is the velocity of the exhaust gas after it leaves the rocket, V_{rocket} is the velocity of the rocket, and V_{e} is the exhaust velocity of
the rocket, which is part of the rocket’s specification.
Combining these relations yields:
This is the thrust
equation – a fundamental principle in studying rockets which provides the
thrust (net propulsive force) acting on a rocket. Let’s look at each term.
The momentum term
describes the rate at which momentum is being ejected from the rocket through
the exhaust.
The pressure term
describes how pressure forces act on the base of the rocket – note that it can
be either positive or negative. As the
rocket moves through the atmosphere, the ambient pressure changes; on the
ground, the pressure term is negative and decreases engine performance; at high
altitudes and in space, there is no ambient pressure so the higher pressure of
the rocket exhaust provides a boost.
You can see how even starting with a basic
principle (like Newton’s second law of motion) and a freebody diagram, we can
derive a useful equation that helps us study real physical situations – like a
rocket! The thrust equation provides a
useful estimate of how much force a rocket will be able to generate. Engineers use the thrust equation when
designing rockets to understand how the rocket will accelerate and how much
weight it will be able to carry.
Sixty Years Ago in the Space Race:
August 8, 1957: The Americans launched its
third and final suborbital JupiterC rocket to an altitude of 285 miles to
test a reentry vehicle.
