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The quantities v(t) and a(t) are important in physics because these mathematically defined quantities appear in experimentally discovered physical laws. The acceleration is related to the force applied via F=ma, and the velocity determines things like the Doppler Shift and the pressure measured in a Pitot tube - those little bent tubes that stick out of fuselage of an airplane and are bent to face into the airflow so that a pressure-measuring instrument can determine the plane's airspeed.
In our hierarchy of models there are two mutually exclusive special cases: Simple Harmonic Motion caused by a restoring force that varies linearly with the particle's displacement from some center of force, and Motion with Constant Acceleration, caused by a constant force. Motion with Constant Velocity is a special case of Motion with Constant Acceleration specified by the constraint a(t) = 0.
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