Introduction to Sports Biomechanics: Analysing Human Movement ...

Impulse of a force
CAUSES OF MOVEMENT – FORCES AND TORQUES
Newton’s second law of linear motion (the law of momentum) can be expressed symbolically
at any time, t, as F = dp/dt = d(m v)/dt. That is, F, the net external force acting
on the body, equals the rate of change (d/dt) of momentum (p = m v). For an object of
constant mass (m), this becomes: F = dp/dt = m dv/dt = m a, where v is velocity and a is
acceleration. If we now sum these symbolic equations over a time interval we can write:
∫F dt = ∫d(m v); this equals m∫dv, if m is constant. The symbol ∫ is called an integral,
which is basically the summing of instantaneous forces. The left side, ∫F dt, of this
equation is the impulse of the force, for which the SI unit is newton-seconds, N s. This
impulse equals the change of momentum of the object (∫d(m v) or m∫dv if m is
constant). This equation is known as the impulse–momentum equation and, with its
equivalent form for rotation, is an important foundation of studies of human dynamics
in sport. The impulse is the area under the force–time curve over the time interval of
interest and can be calculated graphically or numerically. The impulse of force can be
found from a force–time pattern, such as Figure 5.2 or 5.12, which is easily obtained
from a force platform.
The impulse–momentum equation can be rewritten for an object of constant mass
(m) as F ∆t = m ∆v, where F is the mean value of the force acting during a time interval
∆t during which the speed of the object changes by ∆v (the Greek symbol delta, ∆,
simply designates a change). The change in the horizontal velocity of a sprinter from
the gun firing to leaving the blocks depends on the horizontal impulse of the force
exerted by the sprinter on the blocks (from the second law of linear motion) and is
inversely proportional to the mass of the sprinter. In turn, the impulse of the force
exerted by the blocks on the sprinter is equal in magnitude but opposite in direction to
that exerted, by muscular action, by the sprinter on the blocks (from the third law of
linear motion). Obviously, a large horizontal velocity off the blocks is desirable. However,
a compromise is needed as the time spent in achieving the required impulse (∆t)
adds to the time spent running after leaving the blocks to give the recorded race time.
The production of a large impulse of force is also important in many sports techniques
of hitting, kicking and throwing to maximise the speed of the object involved. In javelin
throwing, for example, the release speed of the javelin depends on the impulse applied
to the javelin by the thrower during the delivery stride and the impulse applied by
ground reaction and gravity forces to the combined thrower–javelin system throughout
the preceding phases of the throw. In catching a ball, the impulse required to stop the
ball is determined by the mass (m) and change in speed (∆v) of the ball. The catcher
can reduce the mean force (F) acting on his or her hands by increasing the duration of
the contact time (∆t) by ‘giving’ with the ball.
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