Differences

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

--- 183_notes:examples:positionpredict [2014/07/10 19:44] – created caballero
+++ 183_notes:examples:positionpredict [2024/01/31 16:37] (current) – [Setup] caballero
@@ Line 3: / Line 3: @@
 ===== 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). 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.
+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.