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Constant speed
Storyboard

To describe how the position changes over time, one must study its variation through time.

The relationship between the change in position is equivalent to the distance traveled over the elapsed time, which, when divided by that time, becomes the speed.

For a finite elapsed time, the speed corresponds to the average speed during that interval.

Note: Velocity encompasses speed while also incorporating directional information.

>Model

ID:(608, 0)


Position
Image

The position ($s$) of an object in a one-dimensional system refers to the location of the object in relation to a reference point. This location is expressed as the distance between the object and the origin point. This distance can be a straight line on a Cartesian axis, or it can follow a curved path:

ID:(15, 0)


Initial position
Note

The starting position ($s_0$) is the starting location of an object before any motion begins. This location is defined as the distance between the object and the origin point. This distance can be a straight line on a Cartesian axis or it can follow a curved path.

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Distance traveled
Quote

The distance traveled in a time ($\Delta s$) by an object is measured by measuring the distance between two specific points along a trajectory. This trajectory can be a straight line on a Cartesian axis or a curved path. The distance is calculated by measuring the length of the trajectory between the two starting and ending points.

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Velocity as slope of position curve
Audio

If displacement is graphed as a line between the origin O and point A:

it can be seen that a path has been traveled over a period of time. Therefore, the slope of the graph of path vs elapsed time corresponds to velocity.

If the slope is steeper, it means that a path is covered in less time, which corresponds to a higher velocity.

If the slope is flatter, it means that a path is covered in more time, which corresponds to a lower velocity.

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Time path diagram with horizontal segment
Video

A second type of case is horizontal segments in the distance vs time graph:

If we observe segment AB, we will notice that despite the elapsed time, the distance has not changed. This means that the object is at rest. Therefore, horizontal segments, which correspond to zero slope, correspond to periods when the velocity is zero.

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Time path graph for constant speed and initial time
Unit

For the case of constant velocity and initial time, the position can be calculated using the values the position ($s$), the starting position ($s_0$), the constant velocity ($v_0$), the time ($t$), and the start Time ($t_0$) with the following equation:

$ s = s_0 + v_0 ( t - t_0 )$



which corresponds to a straight line with:

• a slope equal to the constant velocity ($v_0$)
• a y-intercept at the starting position ($s_0$) for the start Time ($t_0$)

as illustrated below:

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Negative slope
Code

When a segment on a graph has a negative slope, as shown below:

it represents a situation in which the object returned from position B to C, which is a distance of zero from the origin. In other words, negative slopes correspond to traveling in the opposite direction, not moving away but getting closer to the origin.

ID:(2245, 0)


Paradox of the body at rest
Flux

When a body is said to be "at rest", it means that it is at rest relative to our reference frame or coordinate system. However, this "rest" is entirely relative, meaning that from a body that moves relative to our system, the body at "rest" is also in motion.

In this sense, there is no such thing as an "absolute rest", it only exists as something relative to a particular reference frame. Therefore, in general, all velocity measurements are relative to a particular reference frame.

For example, if a body appears to move very slowly, it only means that its velocity is very similar to the velocity of the reference frame in which the slow movement is observed.

ID:(4405, 0)


Constant velocity
Description

To describe how position evolves over time, it is necessary to analyze its variation throughout time. The relationship between the variation in position is equivalent to the distance traveled over the elapsed time, which, when divided by that time, becomes the velocity. The definition of velocity assumes that the change in position and time occurs uniformly. If this is not the case, the velocity defined in this way corresponds to the average velocity over that time interval.

Variables
Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$v_0$
v_0
Constant velocity
m/s
$\Delta s$
Ds
Distance traveled in a time
m
$\bar{v}$
v_m
Mean Speed
m/s
$s$
s
Position
m
$t_0$
t_0
Start Time
s
$s_0$
s_0
Starting position
m
$t$
t
Time
s
$\Delta t$
Dt
Time elapsed
s

Calculations

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Equation
Solved
Translated

Calculations

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Equation
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Equations

With the distance traveled in a time ($\Delta s$) it is with the position ($s$) and the starting position ($s_0$):

$ \Delta s = s - s_0 $



and the time elapsed ($\Delta t$) is with the time ($t$) and the start Time ($t_0$):

$ \Delta t \equiv t - t_0 $



The equation for average velocity:

$ v_0 \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$



can be written as:

$v_0 = \bar{v} = \displaystyle\frac{\Delta s}{\Delta t} = \displaystyle\frac{s - s_0}{t - t_0}$



thus, solving for it we get:

$ s = s_0 + v_0 ( t - t_0 )$

(ID 3154)

If we start from the starting position ($s_0$) and want to calculate the distance traveled in a time ($\Delta s$), we need to define a value for the position ($s$).

In a one-dimensional system, the distance traveled in a time ($\Delta s$) is simply obtained by subtracting the starting position ($s_0$) from the position ($s$), resulting in:

$ \Delta s = s - s_0 $

(ID 4352)

Examples

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