Velocity and Acceleration 33it is clear that the tangent vector 2 points along the fuselage from the tail
to the nose, and the normal vector p points up perpendicular to the wings
(draw a picture!) In this construction, the vector b points along the wings
to make u,p,b a right-handed triple. The center of mass of the plane is
the natural common origin of the three vectors. The rolling of the plane,
the rotation around u, lifts one side of the plane relative to the other and
is controlled by the ailerons on the back edges of the wings. Yawing, the
rotation around p, moves the nose left and right and is controlled by the
rudder on the vertical part of the tail. Pitching of the plane, the rotation
around 6, moves the nose up and down and is controlled by the elevators
on the horizontal part of the tail.
Note that rolling and pitching are the main causes of motion sickness.
1.3.4 Velocity and Acceleration
Let the curve C, defined by the vector function r = r(t), be the trajectory
of a point mass in some frame O. Between times t and t + At the point
moves through the arc length As — s(t + At) — s(t), and therefore ds(t)/dt
is the speed of the point along C. As we derived on page 30,
dr ds dr ds ^. , ._ „ _„,
— = —— = —«(<). (1.3.18) dt dtds dt v ' K 'Therefore, we define the velocity v(t) as
v{t) = dr/dt.In particular, ||u|| = \ds/dt\ = ds/dt, that is, the speed is the magnitude of
the velocity; recall that the arc length s — s(t) is a non-decreasing function
of t. This mathematical definition of velocity agrees with our physical
intuition of speed in the direction of the tangent line, while making the
physical concept of velocity precise, as required in a quantitative science.
The definition also works well in practical problems of motion. Indeed,
precise physics is mathematical physics.
Similarly, the acceleration a(t) of the point mass is, by definition,
a(t)=v'(t) = r"(t).
Since dvjdt = d((ds/dt)u(t))/dt, the product rule (1.3.4) implies
dv _ d^2 s _ ds du ds
~dl ~ ~dl?U^ ' + ~di ~ds"di