1.3.1 Applications of calculus
Let’s see how this relates to calculus. If an object is moving in
one dimension, we can describe its position with a functionx(t). The
derivativev= dx/dtis called the velocity, and the second derivative
a= dv/dt= d^2 x/dt^2 is the acceleration. Galilean relativity tells us
that there is no detectable effect due to an object’s absolute velocity,
since in some other frame of reference, the object’s velocity might
be zero. However, an acceleration does have physical consequences.i/This Air Force doctor volun-
teered to ride a rocket sled as a
medical experiment. The obvious
effects on his head and face are
not because of the sled’s speed
but because of its rapidchanges
in speed: increasing in (ii) and
(iii), and decreasing in (v) and (vi).
In (iv) his speed is greatest, but
because his speed is not increas-
ing or decreasing very much at
this moment, there is little effect
on him.(U.S. Air Force)Observers in different inertial frames of reference will disagree
on velocities, but agree on accelerations. Let’s keep it simple by
continuing to work in one dimension. One frame of reference uses a
coordinate systemx 1 , and the other we labelx 2. If the positivex 1
andx 2 axes point in the same direction, then in general two inertial
frames could be related by an equation of the formx 2 =x 1 +b+ut,
whereuis the constant velocity of one frame relative to the other,
and the constantbtells us how far apart the origins of the two
coordinate systems were att= 0. The velocities are different in the
two frames of reference:
dx 2
dt
=
dx 1
dt
+u,Suppose, for example, frame 1 is defined from the sidewalk, and
frame 2 is fixed to a float in a parade that is moving to our left at
a velocityu = 1 m/s. A dog that is moving to the right with a
velocityv 1 = dx 1 /dt= 3 m/s in the sidewalk’s frame will appear
to be moving at a velocity ofv 2 = dx 2 /dt= dx 1 /dt+u= 4 m/s in
the float’s frame.Section 1.3 Galilean relativity 67