1. Understand the itinerary equations: Recognize and understand the equations of motion, especially for the \hat{x} axis, and their application in describing the motion of bodies.
2. Apply the equations to rectilinear uniform motion: Learn to simplify the itinerary equations when acceleration is zero, to describe rectilinear uniform motion.
3. Identify the characteristics of rectilinear uniform motion: Know that in this type of motion, the direction and speed of the object remain constant.
4. Calculate the position of a mobile: Use the equations of motion to determine the position of an object at any time, given its initial position and speed.

The Itinerary Equations of R.U.M.

In the previous class we reviewed the reasoning that leads to the itinerary equations for describing the motion of bodies. Now it’s time to put them in the context of some simple phenomena to start practicing. The simplest of these phenomena corresponds to rectilinear uniform motion.

Before continuing, let’s recall the form of the itinerary equations. For the coordinate axis \hat{x} these are:

\begin{array}{rcl} a_x (t) & = & a_{0x} \\ \\ v_x(t) & = & a_{0x}t+v_{0x}\\ \\ x(t) & = & \frac{1}{2}a_{0x}t^2 + v_{0x}t + x_0 \end{array}

Here, a_{0x}, v_{0x}, and x_0 represent the acceleration, speed, and initial position of the mobile, a_x(t), v_x(t), and x(t) are the acceleration, speed, and position of the mobile over time, respectively. All these expressions are written similarly for the other coordinate axes \hat{y} and \hat{z} to represent movement in 1, 2, and 3 spatial dimensions.

Contextualizing Rectilinear Uniform Motion

To place the itinerary equations in the context of rectilinear uniform motion, we must answer the question: What is rectilinear uniform motion? and transform that question into conditions that we can impose on the itinerary equations:

• The direction of motion does not change over time
• nor does the speed of motion change

All this is summarized by establishing that the acceleration in the three axes is zero at all times; with this, the itinerary equations are as follows:

\begin{array}{rcl} v_x(t) & = & v_{0x}\\ \\ x(t) & = & v_{0x}t + x_0 \end{array}

And similarly for the other axes. Knowing the initial position and speed, it is possible to determine the position of the mobile at any time.

Using these equations and starting from initial conditions for the position and speed, it is possible to determine the position of the mobile at any time.

Example Exercises:

1. Find the position at the instant t=15[s] of a mobile with the initial position and speed given below:
   1. v_{0x}=0.25[m/s]\;\;x_0=0.3[m]
   2. v_{0x}=10[km/h]\;\;x_0=15[m]
   3. v_{0x}=-3[m/min]\;\;x_0=4[in]
2. Two trains are initially separated by a distance of 10 kilometers. Both move along the same track; the first with speed v_{1}=20[km/h], and the second with speed v_2=15[km/h] but in the opposite direction.
   1. How long do the trains take to collide?
   2. What distance does each train cover from its initial position to the point of impact?