Investigation 4B: A model for accelerated motion

Essential questionsHow can a velocity versus time graph be used to determine displacement?
An object’s position changes as it accelerates. Just like a moving object’s position versus time graph can communicate information about its velocity and acceleration, graphs of velocity can communicate information about how an accelerating object’s position is changing, given its initial velocity and position.
Part 1: Determining displacement from motion graphs

  1. Open the 04B_ModelsAcceleratedMotion experiment file in your software, and then connect your Smart Cart using Bluetooth.
  2. Set up your equipment like the picture with the plunger on the cart facing the end stop.
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  1. Push-in the cart’s plunger two clicks, and then set the cart at the bottom of the track with the plunger resting against the end stop.
  2. Start recording data, and then launch the cart up the track. Stop recording data just before the cart starts rolling back down the track.
  1. Sketch a copy of your graph and indicate which curve is velocity and which is position.
  2. How can you tell from your graph that the cart was accelerating?
  3. Use the area tool in your software to find the area under your velocity time graph.
  4. Find the value of the cart’s position when the last velocity data point was recorded. 
  5. If xi = 0 m at the bottom of the track, what was the cart’s total displacement Δx? How does the cart’s total displacement compare to the area under its velocity-time graph?
Part 2: Extending to other graphs

  1. Hide your Part 1 data so the graph is blank.
  2. Repeat the same Part 1 procedure, except this time, after you launch the cart, let the cart bounce off the end stop several times. Record data for about 8 seconds as the cart bounces.
  1. Sketch a copy of your new graph and indicate which curve is velocity and which is position.
  2. Find the area under the new velocity time graph, and the total displacement of the cart.
  3. How does the cart’s total displacement compare to the area under its velocity time graph?
  4. Based on your results in Parts 1 and 2, what is the relationship between the area under an object’s velocity time graph and the object’s displacement?
  5. The cart’s motion in Parts 1 and 2 was mostly uniform (constant acceleration). Does the relationship between the area under an object’s velocity time graph and the object’s displacement extend to more erratic/random motion (variable acceleration)? Test the relationship by recording a third run of erratic/random velocity data, rolling the cart back-and-forth with your hand. Explain your results.
The timer utility in the center panel of this interactive tool can be used to generate a sound every 1.0 seconds—or any time interval of your choosing (down to 0.3 s). This may be used in this lesson for a class demonstration or activity.

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