Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 38

changes in motion. This can be the inertia of a stationary
object or that of a steadily moving object to any accelera-
tion. In many relevant physical situations the mass of a
given volume element is constant with time and therefore
it is the velocity term that determines the conservation of
momentum . When velocity changes in magnitude or direc-
tion, momentum changes. As we shall see a little later, any
such change in momentum over time is due to an equiva-
lent force, F

magma that is losing pressure and exsolving gas in
bubbles, and so on. The example of colliding solid bodies
such as sand grains violently impacting on a desert floor or
colliding in a granular fluid brings us to one definition of
the conservation of linear momentum for such solid-solid
interactions: “. . . the sum of momenta of an isolated
system of two bodies that exert forces on one another is a
constant, no matter what form the forces take ....”
In other words, the collision of bodies or their interaction
leads to no change in overall energy (the production of
collisional heat energy is included in the balance). This
principle of the conservation of momentum forms the
basis of Newton's Third Law (see Section 3.3).

d p /d t . We expect momentum changes to
arise in Nature very frequently: in fluids when an air or
water mass changes direction and/or speed due to changes
in external conditions; in fluid flow over solid boundaries
where a velocity gradient is set up; in the ascent of molten

3.2 Acceleration

3.2.1 A simple introduction

Object 1

Object 2

u proportional t 1.5

u proportional t
5

Acceleration is a very obvious physical phenomenon; we
feel it driving or being driven. In both cases the effect is
due to a change of velocity; the more sudden the change,
the more reaction we feel. The very fact that a body can
“feel” acceleration and is forced to move in response to it
means that the phenomenon is somehow connected to
that of force. Another kind of universal acceleration con-
cerns a falling body through a frictionless medium, such as
a solid through a vacuum, or through air whose resistance
to motion is low (and may sometimes be neglected). Here
the falling solid is attracted by Earth's gravity field. Some
examples of the use of uniform acceleration appropriate to
such falling bodies are given in Cookie 4. In other more
“resistant” liquids, a steady rate of fall is achieved after an
interval such that the downward acceleration due to grav-
ity is quickly balanced by the resistance of the liquid
medium.
Acceleration (from now on we use the term without
regard for the sign, positive or negative) of the kinds men-
tioned is most simply imagined as change of velocity over
time (Fig. 3.5). Thus in differential form (appendix),
a

5

6.0 6.0

7.0 7.8

8.0 10.2

13

9.0

Speed-time graph

2. Nonlinear
acceleration

1. Linear
acceleration

t 1 t 2 t 3 t 4 t 5

Time, t

d u /d t , with dimensions of LT 2 . The standard accel-
eration due to gravity, g , at sea level is 9.81 ms 2 . We stress
that natural accelerations may be extreme compared with
this; for example, turbulent eddies are subject to accelera-
tions of many times gravity (order

Fig. 3.5 Acceleration. Vectors for W to E motion at velocity, u , for
times t 1 - t 5 .

3.2.2 Complications in moving fluids

10 3 g ) and the
Earth's surface is subject to several g acceleration during
earthquake motions. By way of contrast, a slow-moving
lithospheric plate may change velocity so slowly over such
a long time period (of the order of 10 6

Now we consider constant flow or discharge of fluid
through conduits, channels, cols, or gates when the pas-
sageway has varying cross-sectional area along its length
(Fig. 3.6). There is no change of velocity over time at any

years) that the

acceleration can be practically neglected.