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183_notes:constantf [2021/02/04 23:27] – [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.
  
 +{{youtube>1RP2oSBAQJI}}
  
-==== Predicting the Motion ==== + 
 + 
 +====== Predicting the Motion ======
  
 Consider a fan cart that is released on a low-friction track. Here's a video of the situation. Consider a fan cart that is released on a low-friction track. Here's a video of the situation.
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 \\ \\
-=== Deriving the Equation for Constant Force Motion in 1D ===+===== Deriving the Equation for Constant Force Motion in 1D ====
  
 If you choose the horizontal direction to be the x-direction, we have the following equations to describe the motion. If you choose the horizontal direction to be the x-direction, we have the following equations to describe the motion.
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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.
 +
  
 \\ \\
  
-== Connection to Energy ==+ 
 + 
 +==== Connection to Energy ====
  
 As you will read, the motion of systems can also be predicted or explained by using the [[183_notes:define_energy|energy principle]] in addition to or, as an alternative, to using the [[183_notes:momentum_principle|momentum principle]]. You will find that using energy, you can often think about [[183_notes:grav_and_spring_pe|the initial and final states of the system's motion]] and not how that motion evolves (e.g., over what time the motion occurs).  As you will read, the motion of systems can also be predicted or explained by using the [[183_notes:define_energy|energy principle]] in addition to or, as an alternative, to using the [[183_notes:momentum_principle|momentum principle]]. You will find that using energy, you can often think about [[183_notes:grav_and_spring_pe|the initial and final states of the system's motion]] and not how that motion evolves (e.g., over what time the motion occurs). 
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 Again, as you will read, this equation can also be derived from [[183_notes:work|the relationship between kinetic energy and work]]. Again, as you will read, this equation can also be derived from [[183_notes:work|the relationship between kinetic energy and work]].
  
-=== Constant Force in 3D ===+\\ 
 +==== Summary of Constant Force  ==== 
 + 
 +The relationship between force and acceleration (even for a variable net force): $\vec{F}_{net}=m\vec{a}$ OR $\vec{a}=\frac{\vec{F}_{net}}{m}$. 
 + 
 +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: 
 +$$x_{f} = x_{i} + v_{avg,x} \Delta t$$ 
 +$$v_{fx} = v_{ix} + \dfrac{F_{net,x}}{m} \Delta t$$ 
 +$$v_{avg,x} = \dfrac{v_{ix} + v_{fx}}{2} = v_{ix}+ \dfrac{1}{2}\dfrac{F_{net,x}}{m} \Delta t $$ 
 +$$x_{f} =  x_{i} + v_{ix} \Delta t + \dfrac{1}{2}\dfrac{F_{net,x}}{m} \Delta t^2$$ 
 +$$v_{xf}^2 = v_{xi}^2 + 2\dfrac{F_{net,x}}{m}\Delta x$$  
 + 
 +\\ 
 + 
 +==== Constant Force in 3D ====
  
 The derivation for each dimension is similar (so long as the force is constant in each direction). The result is the following general equation, The derivation for each dimension is similar (so long as the force is constant in each direction). The result is the following general equation,
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