183_notes:constantf

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183_notes:constantf [2021/02/18 20:52] – [Predicting the Motion] stumptyl183_notes:constantf [2023/01/13 19:59] (current) – hallstein
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 Depending on how you select your coordinate system, it might mean that more than one component of the momentum vector changes. Often, it is convenient to select a coordinate system where the net force is aligned with a coordinate direction, then only one momentum vector component changes in time. Depending on how you select your coordinate system, it might mean that more than one component of the momentum vector changes. Often, it is convenient to select a coordinate system where the net force is aligned with a coordinate direction, then only one momentum vector component changes in time.
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 +{{youtube>1RP2oSBAQJI}}
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 From this equation, you can determine the arithmetic average velocity, which in this case is equal to the average velocity. From this equation, you can determine the arithmetic average velocity, which in this case is equal to the average velocity.
- +$$v_{avg,x} = \dfrac{v_{ix} + v_{fx}}{2} = \dfrac{ v_{ix} + v_{ix} + \dfrac{F_{net,x}}{m} \Delta t}{2} 
-$$v_{avg,x} = \dfrac{v_{xi} + v_{xf}}{2} = \dfrac{ v_{xi} + v_{ix} + \dfrac{F_{net,x}}{m} \Delta t}{2} + = \dfrac{2v_{ix}}{2}+ \dfrac{\dfrac{F_{net,x}}{m} \Delta t}{2} = v_{ix}+ \dfrac{1}{2}\dfrac{F_{net,x}}{m} \Delta t $$
- = \dfrac{2v_{xi}}{2}+ \dfrac{\dfrac{F_{net,x}}{m} \Delta t}{2} = v_{xi}+ \dfrac{1}{2}\dfrac{F_{net,x}}{m} \Delta t $$+
  
 By using this average velocity in the [[183_notes:displacement_and_velocity|position update formula]], you obtain the final expression that predicts the location of the system given only information about its //initial position, velocity, and the force acting on it.//  By using this average velocity in the [[183_notes:displacement_and_velocity|position update formula]], you obtain the final expression that predicts the location of the system given only information about its //initial position, velocity, and the force acting on it.// 
  
-$$x_{f} = x_{i} + v_{avg,x} \Delta t = x_{i} + v_{xi} \Delta t + \dfrac{1}{2}\dfrac{F_{net,x}}{m} \Delta t^2$$+$$x_{f} = x_{i} + v_{avg,x} \Delta t = x_{i} + v_{ix} \Delta t + \dfrac{1}{2}\dfrac{F_{net,x}}{m} \Delta t^2$$
  
 In physics, the information about the system prior to predicting its motion is called the "initial state" of the system. The starting values of these properties (position, velocity, net force) are called the "initial conditions" of the system. In physics, the information about the system prior to predicting its motion is called the "initial state" of the system. The starting values of these properties (position, velocity, net force) are called the "initial conditions" of the system.
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 ==== Connection to Energy ==== ==== Connection to Energy ====
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 +==== Summary of Constant Force  ====
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 +The relationship between force and acceleration (even for a variable net force): $\vec{F}_{net}=m\vec{a}$ OR $\vec{a}=\frac{\vec{F}_{net}}{m}$.
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 +The following 1D equations are valid ONLY if the net force (and therefore, the acceleration) is constant.  These equations are commonly known as kinematic equations:
 +$$x_{f} = x_{i} + v_{avg,x} \Delta t$$
 +$$v_{fx} = v_{ix} + \dfrac{F_{net,x}}{m} \Delta t$$
 +$$v_{avg,x} = \dfrac{v_{ix} + v_{fx}}{2} = v_{ix}+ \dfrac{1}{2}\dfrac{F_{net,x}}{m} \Delta t $$
 +$$x_{f} =  x_{i} + v_{ix} \Delta t + \dfrac{1}{2}\dfrac{F_{net,x}}{m} \Delta t^2$$
 +$$v_{xf}^2 = v_{xi}^2 + 2\dfrac{F_{net,x}}{m}\Delta x$$ 
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 +\\
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 ==== Constant Force in 3D ==== ==== Constant Force in 3D ====
  
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