MIT 8.01 Lesson 4: 1-D Motion - General
Lesson Summary
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We consider the motion of a point particle in one dimension, which can occur either because the particle is constrained to move along a track, or because we restrict attention of one Cartesian component of a particle's motion. |
In general, the force applied to this particle can vary arbitrarily with time. Hence the particle will have time-varying acceleration in proportion, a(t). The resulting motion may then be found using calculus: the velocity v(t) is the integral of the acceleration a(t) plus the velocity at the start of the interval of integration. Mathematically we'd say that the arbitrary constant of integration is constrained by the initial condition on the velocity. To get x(t) we integrate v(t) and add the initial position. Learning Objectives
By the end of this Lesson, you should be able to:
- Read and understand the One-Dimensional Motion (General) model summary.
- Accomplish the Learning Objectives listed in that summary.
- Summarize the structure of the Hierarchy of models that fall under the One-Dimensional Motion (General) model, and state the specializations required for each sub-model to apply.
Introduction
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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