This page is intended for college students and high school students
who are studying calculus.
For younger students, a simpler explanation of the information on this page is
available on the
For high school and middle school students there is
another version without calculus.
Sir Isaac Newton first presented his three laws of motion
in the "Principia Mathematica Philosophiae Naturalis" in 1686. His second law
defines a force to be equal to the change in momentum
with a change in time.
Momentum is defined to be the mass m of an object times its velocity
Let us assume that we have an airplane at a point "0" defined by its
location X0 and time t0. The airplane has
a mass m0 and travels at velocity V0.
The airplane is subjected to an external force F and moves to a point "1",
which is described by a new location X1 and time t1. The mass and
velocity of the airplane change during the flight to values m1 and V1.
Newton's second law can help us determine the new values of V1 and m1,
if we know how big the force F is.
Using calculus to describe Newton's second law:
F = d (m * V) / dt
This differential equation can be solved with the boundary conditions that
we described above assuming that we know the variation of the force F as a
function of time.
Let us assume that the mass stays a constant value equal to m.
This assumption is pretty good for an airplane, the only change in mass would be for the
fuel burned between point "1" and point "0". The weight of the fuel is probably
small relative to the weight of the rest of the airplane, especially if we only
look at small changes in time.. If we were discussing the
flight of a
then certainly the mass remains a constant. But if we were discussing the flight of a
then the mass does not remain a constant and we would have to specify how the mass varies
with time to perform the integration.
For a constant mass m, Newton's second law looks like:
F = m * dv / dt
The derivative of velocity with respect to time is the definition of the acceleration a.
The second law then reduces to the more familiar product
of a mass and an acceleration:
F = m * a
Remember that this relation is only good for objects that have a constant mass.
This equation tells us that an object subjected to an external force will accelerate and that the
amount of the acceleration is proportional to the size of the force. The amount of acceleration
is also inversely proportional to the mass of the object; for equal forces, a heavier object will
experience less acceeration than a lighter object. Considering the momentum equation, a force
causes a change in velocity; and likewise, a change in velocity generates
a force. The equation works both ways.
The velocity, force, acceleration, and
momentum have both a magnitude and a direction associated with them.
Scientists and mathematicians call this a
The equations shown here are actually vector equations and
can be applied in each of the
component directions. We have only looked at one direction,
and, in general, an object moves in all three directions (up-down, left-right,
The motion of an aircraft resulting from
aerodynamic forces, aircraft
weight, and thrust
can be computed by using the second law of motion. But there is a fundamental problem
when dealing with aerodynamic forces. The aerodynamic forces depend on the
square of the velocity. So integrating the differential equations
becomes a bit more complicated. We show the details of the integration at the web page on
flight equations with drag.
Newton's Laws of Motion:
Forces, Torques and Motion:
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