2009

■ We assume that in the time period h all the external forces acting on the
ball are constant, so air resistance and wind and gravity do not change
during this time.
■ We assume that the math we use to calculate the new position is accurate
In general, the first assumption is usually valid, except at relativistic or
quantum scales, which we can assume should be handled by other systems.
The remaining two, however, cause problems and are closely linked to the
time period h over which we’re performing the calculations. We’ll now
examine the effect of the size of this time period on the accuracy of the
simulation.
Time Steps
In general, the forces acting on an object are rarely truly constant; gravity is
close to being constant, but most other forces like wind and air resistance are
not. So, taking the cannon ball example, imagine a windy layer in the
atmosphere that the cannon ball passes through, as shown in the next figure.
In the simulation on the left we assume we’re taking steps of one second; this
is actually a relatively large interval for a physics simulation, but is used here
to illustrate the point. We know all the forces acting on the ball at time t1 so
we use some math to predict the new position and velocity at time t2, after
one second has elapsed. During this period, we assume that the wind force
acting on the ball is constant. In this example, we’ll calculate the new position
above the region of high wind, so we’ll effectively have missed the windy bit
by taking too great a jump. In the second example on the right, we’re using
time steps of ½ second. In this case, after determining the new position at
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