183_notes:graphing_motion

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision		 Previous revision
183_notes:graphing_motion [2025/11/12 21:12] – hallstein183_notes:graphing_motion [2026/01/04 20:13] (current) – hallstein
Line 13: Line 13:
 $$\vec{r}_f=\vec{r}_i + \vec{v}_{avg} \Delta t$$ $$\vec{r}_f=\vec{r}_i + \vec{v}_{avg} \Delta t$$
  
-While the motion of the car, in principle, can occur 3 dimensions, it's not possible to represent all three dimensions and the time variable on a single 2-D graph. So, we have to select a component of the car's position (or velocity) to plot. In this case, let's assume the car moves to the right (i.e., in the +x direction). Perhaps, the plot of the car's position vs time looks like the plot linked here: [[https://msuperl.org/interactive/mechanics/CV_position_vs_time.html|Constant velocity position vs. time]] +While the motion of the car, in principle, can occur 3 dimensions, it's not possible to represent all three dimensions and the time variable on a single 2-D graph. So, we have to select a component of the car's position (or velocity) to plot. In this case, let's assume the car moves to the right (i.e., in the +x direction). Perhaps, the plot of the car's position vs time looks like the plot below: 
-/* + 
-{{url>https://chart-studio.plotly.com/~PERLatMSU/10/graph-of-x-position-vs-time-for-a-car-moving-with-constant-velocity/#plot 640px,480px}} +{{url>https://msuperl.org/interactive/mechanics/CV_position_vs_time.html 640px,480px |Constant velocity position vs. time}} 
-*/+
  
  
Line 36: Line 36:
 $$v_{x} = \mathrm{instantaneous\:slope} = \dfrac{dx}{dt}$$ $$v_{x} = \mathrm{instantaneous\:slope} = \dfrac{dx}{dt}$$
  
-For position versus time graphs where the position does not change linearly, you might need to determine (by taking the derivative) or approximate (by measuring very close points) the instantaneous velocity to model or explain the motion. For example in the graph linked here: [[https://msuperl.org/interactive/mechanics/CA_position_vs_time.html|Constant force position vs. time]], a car moves to the right under [[:183_notes:constantf|constant force]]. Here, the slope (and thus, the velocity) changes at a constant rate and the average and instantaneous velocities are not the same.+For position versus time graphs where the position does not change linearly, you might need to determine (by taking the derivative) or approximate (by measuring very close points) the instantaneous velocity to model or explain the motion. For example in the graph below, a car moves to the right under [[:183_notes:constantf|constant force]]. Here, the slope (and thus, the velocity) changes at a constant rate and the average and instantaneous velocities are not the same.
  
-/*{{url>https://chart-studio.plotly.com/~PERLatMSU/15.embed 640px,480px}}*/+{{url>https://msuperl.org/interactive/mechanics/CA_position_vs_time.html 640px,480px |Constant force position vs. time }}
  
  
Line 45: Line 45:
 \\ \\
  
-Sometimes, you will want to graph the velocity of the object as a function of time. Again, you have to graph a single component at a time. So, let's go back to the example of a car moving with constant velocity. In that case, we'd expect the velocity vs time graph to be a flat, horizontal line taking on the value of the slope. In the graph linked here [[https://msuperl.org/interactive/mechanics/CV_velocity_vs_time.html|Constant velocity, velocity vs. time]], we find that is the case. +Sometimes, you will want to graph the velocity of the object as a function of time. Again, you have to graph a single component at a time. So, let's go back to the example of a car moving with constant velocity. In that case, we'd expect the velocity vs time graph to be a flat, horizontal line taking on the value of the slope. In the graph below, we find that is the case.
- +
-/*{{url>https://chart-studio.plotly.com/~PERLatMSU/15.embed 640px,480px}}*/+
  
-/*{{url>https://chart-studio.plotly.com/~PERLatMSU/15.embed 640px,480px | Constant Velocity (velocity vs time)}}*/+{{url>https://msuperl.org/interactive/mechanics/CV_velocity_vs_time.html 640px,480px |Constant velocity, velocity vs. time}}
  
 In addition, we can use the position update formula to show that the x-displacement ($\Delta x$) is the area under this curve: In addition, we can use the position update formula to show that the x-displacement ($\Delta x$) is the area under this curve:
Line 57: Line 55:
 This is precisely how one defines a [[http://en.wikipedia.org/wiki/Riemann_sum|Riemann sum]] to determine the area under a function. This area is highlighted in light blue, and is precisely equal to the difference in the initial and final position in the first graph (120 m). The car travels at 12 m/s for 10 s. Notice the displacement is positive because the area under the curve is measured from the function to y=0. This is precisely how one defines a [[http://en.wikipedia.org/wiki/Riemann_sum|Riemann sum]] to determine the area under a function. This area is highlighted in light blue, and is precisely equal to the difference in the initial and final position in the first graph (120 m). The car travels at 12 m/s for 10 s. Notice the displacement is positive because the area under the curve is measured from the function to y=0.
  
-For situations where the object does no move with constant velocity, the area under the velocity vs time graph is still the displacement, it just might be slightly more complicated to calculate. For example, here's the velocity vs time graph for when the car moves under a [[:183_notes:constantf|constant force]].  +For situations where the object does no move with constant velocity, the area under the velocity vs time graph is still the displacement, it just might be slightly more complicated to calculate. For example, the graph below is the velocity vs time graph for when the car moves under a [[:183_notes:constantf|constant force]].  
  
-{{url>https://plot.ly/~PERLatMSU/16/640/480 640px,480px | Constant force (velocity vs time)}}+{{url>https://msuperl.org/interactive/mechanics/CA_velocity_vs_time_area_fill.html 640px,480px | Constant force (velocity vs time)}}
  
 The triangular area (highlighted in light blue) under the curve is the displacement of the car in the x-direction. Notice it's positive because it's above the y-axis. "Area under the curve" actually refers the the area between the function and y=0. If the plot is below the y=0, then that part of the area is negative. The triangular area (highlighted in light blue) under the curve is the displacement of the car in the x-direction. Notice it's positive because it's above the y-axis. "Area under the curve" actually refers the the area between the function and y=0. If the plot is below the y=0, then that part of the area is negative.
  • 183_notes/graphing_motion.1762981945.txt.gz
  • Last modified: 2025/11/12 21:12
  • by hallstein