Alexander Fufaev
My name is Alexander FufaeV and here I will explain the following topic:

Uniform (unaccelerated) Motion in Physics


Formula: Unaccelerated Motion
The Change of Position of a Body During an Unaccelerated Movement
What do the formula symbols mean?


Current position \(x\) of a body at time \(t\).


Constant (unchangeable) speed of the body.

Initial position

The current position \( x(0) = x_0 \) of the body at time \(t=0\).


An arbitrary time at which the position \(x(t)\) of the body is calculated.
Table of contents
  1. Formula
  2. Distance traveled
  3. Uniform motion illustrated in a diagram


Imagine you are standing on the side of the road watching an object, for example a car, pass you at a constant speed on a straight road. With physics you can easily predict WHERE the object will be, after 10 seconds, 20 seconds or 100 seconds. For this purpose there is the following law:

Formula anchor Die Änderung der Position eines Körpers bei einer unbeschleunigte Bewegung
The Change of Position of a Body During an Unaccelerated Movement
  • You started to observe the car at the position \(x_0\). \(x_0\) is the initial position of the object. Usually the initial position is set to zero: \(x_0 = 0\).

  • With a velocimeter you found out that the object moves with a constant velocity \( \class{blue}{v} \). For example with 10 meters per second: \( \class{blue}{v} = 10 \, \mathrm{m}/\mathrm{s} \). Every second the object travels 10 meters.

  • \( x(t) \) is the position of the object at the time \( t \). For example, the position \(x\) after 10 seconds we write as \( x(10) \).

Example: What is the position of the vehicle after a certain time?

A vehicle is traveling at 10 meters per second. What is its position \( x \) after 5 seconds? Multiply the velocity \( \class{blue}{v} \) by the time \( t = 5 \, \mathrm{s} \).

Formula anchor

The object is 50 meters away from the initial position x₀ = 0 after 5 seconds.

Note that only if the initial position \(x_0\) is zero, \(x(t)\) yields the distance the object will travel after a given time \( t \). If the initial position is not zero, \(x(t)\) is NOT the distance traveled!

Distance traveled

How do we determine the covered distance of the object within a given time? The covered distance is written as \( \class{red}{\Delta x} \). This is the distance of two positions \( x_1 \) and \( x_2 \) of the object at two different times \( t_1 \) and \( t_2 \).\( x_2 - x_1 \) is the distance covered \( \class{red}{\Delta x} \) by the object within the time \( t_2 - t_1 \). The time difference \( t_2 - t_1 \) is abbreviated as \( \Delta t \).

You get the distance covered \( \class{red}{\Delta x} \), that is the distance between the positions \( x_2 \) and \( x_1 \), by multiplying the speed of the object by the time \( \Delta t \):

Formula anchor Die zurückgelegte Strecke eines Körpers bei einer unbeschleunigten Bewegung
The Distance Traveled by a Body During an Unaccelerated Motion
Example: What distance has the object traveled after 7 seconds?

The object moves at 10 meters per second.

Formula anchor

Uniform motion illustrated in a diagram

You can illustrate the position \(x(t)\) of an object at time \(t\) in a diagram. The position \(x\) is plotted on the vertical axis and the time \(t\) on the horizontal axis. This results in a position-time graph, as shown in Illustration 3. Weg-Zeit-Diagramm bei einer gleichförmigen (unbeschleunigten) Bewegung
Distance-Time Graph During Uniform (Unaccelerated) Motion

The position-time graph during a uniform movement is a straight line. The slope of the straight line is the ratio of the traveled distance \( \class{green}{\Delta x} \) to the time \( \class{brown}{\Delta t} \) required for it. The slope therefore corresponds to the speed of the object! The steeper the line, the faster the object moves.

On the other hand, if you plot the velocity \( v \) on the vertical axis and the time \(t\) on the horizontal axis, you get a velocity-time graph of uniform motion, as shown in Illustration 4. Geschwindigkeit-Zeit-Diagramm bei gleichförmiger (unbeschleunigter) Bewegung
Velocity-Time Graph During Uniform (Unaccelerated) Motion

Since the velocity does NOT change with time in a uniform motion, the result is a horizontal straight line. This means: At any time \(t\), for example at time \(t_1\) or at time \(t_2\) on the horizontal axis, the velocity on the vertical axis always has the same value. This horizontal straight line is a constant velocity function \( \class{blue}{v(}t\class{blue}{)} \).

In the next lesson, we will look at a non-uniform motion, that is, an accelerated motion.