183_notes:examples:positionpredict [Projects & Practices in Physics]

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===== Example: Predicting the location of a object undergoing constant velocity motion =====

A cart is given a slight push along a near frictionless track (as shown in the video below). 

{{ youtube>sdjsaxLfevQ?large }}

After the push, the cart is observed to move with a near constant velocity $\vec{v}_{cart} =\langle 1.2, 0, 0 \rangle \dfrac{m}{s}$. Determine its location after 3 seconds.
==== Setup ====

You need to predict the location of the cart using the information provided and any information that you can collect or assume.

=== Facts ====

  * The cart moves to the right.
  * The cart's velocity is given by $\vec{v}_{cart} =\langle 1.2, 0, 0 \rangle \dfrac{m}{s}$.

=== Lacking ===

  * The initial location of the cart is not known.

=== Approximations & Assumptions ===

  * The interactions of the cart with its surroundings, over the interval that you care about, are negligible. That is, the cart moves with a constant velocity.
  * As a result, the average and instantaneous velocity are equivalent.
  * We will assume the initial location of the cart is $\vec{r}_{i}$.

=== Representations ===

  * The location of the cart can be predicted using the position update formula, $\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t$
  * The motion of the cart is represented using the following motion diagram.
{{url>https://glowscript.org/#/user/danny/folder/Shared/program/FanCarConstantVelocity 660px,420px|Simulation of Fan Cart moving with Constant Velocity}}
 
==== Solution ====

We can compute the final location,

$$\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t = \vec{r}_i + \vec{v}_{cart} \Delta t = \vec{r}_i + \langle 1.2, 0, 0 \rangle \dfrac{m}{s} (3 s) = \vec{r}_i + \langle 3.6, 0, 0 \rangle m$$

You might use the video to define an origin such that the initial position of the cart is $\vec{r}_i = \langle 0.4, 1.1, 0 \rangle m$. With that new information, the final location of the cart can be computed exactly,

$$\vec{r}_f = \vec{r}_i + \langle 3.6, 0, 0 \rangle m = \langle 0.4, 1.1, 0 \rangle m + \langle 3.6, 0, 0 \rangle m = \langle 4.0, 1.1, 0 \rangle m$$.

Notice that $y$-position of the cart remained unchanged because all the motion of the cart was in the $x$-direction.