Environmental Engineering Reference
In-Depth Information
understanding of the frictional forces that act. In such circumstances, the law of
momentum conservation is not so useful.
Momentum conservation will recur frequently throughout the rest of this topic,
not least when we come to discuss collisions in Section 3.3.
2.3.5
Impulse
When forces act over a short time, as in the collision of a ball with the ground,
the detailed time-dependence of the force is usually unknown. We can in some
cases instead work with the time integral of the force, which is equivalent to the
change in momentum. Suppose a force F (t) acts on a particle for a short time,
from t 1 to t 2 . The impulse is defined to be
t 2
t 2
d P
d t
F (t) d t
=
d t
=
P (t 2 )
P (t 1 )
=
P .
(2.31)
t 1
t 1
Example 2.3.7 Estimate the force involved in serving a tennis ball at a speed of
120 kmhr 1 .
Solution 2.3.7 The mass of a tennis ball is roughly 60 g. The ball is hit at the
highest point of the toss where it is stationary, so the change in momentum is
equal to the momentum of the ball after serving. The collision takes typically about
2milliseconds.
10 3
10 3 kg m
×
2 . 0kgms 1 .
P
=
60
×
120
×
=
×
60
60 s
This leads to a mean value of the force:
P
t
2
10 3 N .
F av =
=
N
=
10 3
2
×
This is roughly the weight of a 100 kg mass and that may seem surprising given that
the mass of a typical tennis player is less than 75 kg. To determine what effect this
force has on the tennis player, we can calculate their speed of recoil assuming that
they are off the ground when contact is made:
2 . 0kgms 1
75 kg
P
M
0 . 027 m s 1
v
=
=
or about 0.1 km hr 1 . Thus, despite the large force involved in propelling the ball,
the duration is short enough, and the mass of the player large enough, such that the
player experiences only a very small recoil velocity.
2.3.6 Motion in fluids
When a solid body moves through a fluid, some of the fluid is dragged along
with the body and the fluid acquires momentum in the direction of the object's
motion. This change in momentum of the fluid is associated with a force that the
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