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Figure 1.5. Space curve with position and velocity at time t .
1.6 Rigid-Body Dynamics
1.6.1 Constant Forces
Suppose we have an object in motion in space. For the moment, we will consider
only a particle with position x , or linear motion. If we track this position over
time, we end up with a function x ( t ). In addition, we can consider at a particular
time how fast the object is moving and in what direction. This is the velocity
v ( t ). As the velocity describes how x changes in time, it is also the derivative of
its position, or x . (See Figure 1.5 . )
Assuming that the velocity v is constant, we can create a formula for com-
puting the future position of an object from its current position x 0 and the time
traveled t :
x ( t )= x 0 + v t.
However, most of the time, velocity is not constant, and we need to consider its
derivative, or acceleration a . Assuming a is constant, we can create a similar
formula for v ( t ):
v ( t )= v 0 + a t.
Since velocity is changing at a linear rate, we can substitute the average of the
velocities across our time steps for v in our original equation:
x ( t )= x 0 + t 1
2 ( v 0 + v ( t ))
= x 0 + t 1
2 ( v 0 + v 0 + a t )
= x 0 + v 0 t + 1
2 a t 2 .
Acceleration in turn is derived from a vector quantity known as a force F .
Forces act to push and pull an object around in space. We determine the acceler-
ation from force by using Newton's second law of motion,
F = m a ,
where m is the mass of the object and is constant.
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