Questions: Position, Velocity, Acceleration


One Dimension

In the video below, we see Poli. As Poli moves, a trail of breadcrumbs is laid, one crumb dropped at equally spaced intervals in time.

  1. In the video above, what is the meaning of the cyan (light blue) arrow? [Some possible choices are: position, velocity, acceleration, kinetic energy, angular momentum, or something else.] For this first question, I want you to use your gut. Don’t think about it too much. Just write something down.
  2. Now, I want you to give it a lot of thought. Start with the definition of the response you provided to the previous question, and try to justify your answer rigorously. If your previous choice no longer makes sense, choose a new name for the light blue vector that you can justify. The behavior of the vector should coincide with it’s definition throughout the entire video above.

Now consider this video with both light blue and pink (magenta) vectors.

  1. How are the pink vector and light blue vector related to each other mathematically? Be as specific and rigorous as you can.
  2. Given the mathematical relationship you proposed in the previous question, what is the name of the pink vector?

To answer the next two questions, I want you to return to the first video at the top of this page. You may want to watch the video with your mouse hovering over the pause button so that you can see the elapsed time as you’re watching the dynamics of Poli.

  1. In the first video at the top, the pink vector is not shown. Over what intervals of time, would the pink vector be pointing to the right? To the left? Zero?
  2. Suppose we write the light blue vector as \( \underset{\sim}{\boldsymbol{b}} = b(t) \hat \imath \) and the pink vector as \( \underset{\sim}{\boldsymbol{p}} = p(t) \hat \imath \). Create a plot that shows the scalar components \(b(t)\) and \(p(t)\) as functions of time.

For the next question, consider the video shown below. Now the vectors are not shown. But, Poli does drop the bread crumbs at equal intervals of time.

  1. Based on the “bread crumbs” shown in the video above, draw a plots of \(b(t)\) and \(p(t)\) lined up so that the time axes coincide.

In this last video, the green vector represents the net force acting on Poli. Observe its relationship to the other two vectors.

  1. What did Isaac Newton have to say about the relationship between the green vector and the pink vector? Does the video appear to show the proper relationship between the two vectors? Explain.
  2. If one doubled the mass of Poli, and applied the same time dependent (green) forces, how would the behavior of the pink vector differ?
  3. If one doubled the mass of Poli, and applied the same time dependent (green) force, how would the behavior of the light blue vector differ?

If you haven not figured it out yet, the light blue vector is the velocity of Poli, and the pink vector is acceleration. Answer these last two questions about velocity and acceleration, using instances from the last movie to illustrate your answers.

  1. Suppose that the acceleration vector is zero over some interval of time. What can you say about the velocity over that same interval of time? Find a time interval in the video above which illustrates this.
  2. Suppose that the velocity is zero at some instant of time. Is the acceleration necessarily zero at this instant of time too? Find a time in the video above with illustrates your answer.