Motion quantities and uniform motion

Motion is a part of countless natural phenomena. The Earth revolves around the Sun, people move on the Earth’s surface, and the molecules in the atmosphere are in constant motion. If we want to measure motion, we simply need a pre-determined distance and a stopwatch. We can use the distance and the watch to determine an object’s position at different times and then use the data to calculate different quantities related to motion, such as velocity and acceleration.

Determining an object’s motion requires a reference point that the motion is studied against. The reference point can be a starting line at a running track or a location that you are travelling to, for example. In a coordinate system, the reference point is the origin. Motion changes the position relative to the origin. The change can be presented graphically with a position-time coordinate system. This presentation method is also called a position vs. time graph. In the video below, the researchers use an ultrasonic sensor to measure the motion of a remote-controlled car. The sensor measures the car’s distance relative to the probe. The starting point of the car is used as the origin, and the initial direction of the motion is considered the positive direction.

Velocity, average velocity and speed

Velocity ($v$) represents a change in position relative to time. The average velocity ($v_{\text{av}}$) of an object is the ratio of displacement and the time it took to achieve the displacement. Average velocity is only affected by how great the displacement was and how much time it took, so it does not show possible changes in velocity during the displacement. By calculating the average velocity, we can find out what velocity an object would have moved at if the velocity had remained constant. Like displacement, the average velocity value can be either positive or negative. The positive or negative prefix of velocity value shows the direction of motion relative to the origin. Now, let’s apply what we have learned so far to the remote-controlled car we saw in the video above. We will use the motion graph of the car to determine its displacement and average velocity between two different points in time.

Displacement A from point $x_1$ to $x_2$ occurs from 2.0 s... 4.0 s and is 0.21 m. Therefore, the average velocity is

$v_{\text{A}}={\displaystyle \frac{\mathrm{\Delta <!--\; \Delta \; -->}x}{\mathrm{\Delta <!--\; \Delta \; -->}t}}={\displaystyle \frac{x_2-<!--\; -\; -->x_1}{t_2-<!--\; -\; -->t_1}}={\displaystyle \frac{0.62\text{ m}-<!--\; -\; -->0.41\text{ m}}{4.0\text{ s}-<!--\; -\; -->2.0\text{ s}}}\approx <!--\; \approx \; -->0.11\text{ m/s}$

Displacement B from point $x_2$ to $x_3$ occurs from 4.0 s to 8.0 s and is -0.51. Therefore, the average velocity is

$v_{\text{B}}={\displaystyle \frac{\mathrm{\Delta <!--\; \Delta \; -->}x}{\mathrm{\Delta <!--\; \Delta \; -->}t}}={\displaystyle \frac{\mathrm{\Delta <!--\; \Delta \; -->}{x}_3-<!--\; -\; -->x_2}{t_3-<!--\; -\; -->t_2}}={\displaystyle \frac{0.11\text{ m}-<!--\; -\; -->0.62\text{ m}}{8.0\text{ s}-<!--\; -\; -->4.0\text{ s}}}\approx <!--\; \approx \; -->-<!--\; -\; -->0.13\text{ m/s}$

Velocity is a quantity that represents motion: it accounts for the magnitude and direction of the motion. Speed is a quantity that only represents the magnitude of motion. The speed of a remote-controlled car is always positive regardless of whether the car moves to the left or the right. To calculate the average speed, divide the distance with the time it took to travel: ${\displaystyle \frac{s}{\mathrm{\Delta <!--\; \Delta \; -->}t}}$. Using the same points from above, the average speed of the remote-controlled is 0.11 m/s at point A and 0.13 m/s at point B.

The simplest model of motion is uniform motion. There are many cases of natural motion that can be modelled as uniform motion. Light moves with uniform motion, and so do trains, at least approximately. Uniform motion can also be seen in the video below, where two low-friction carts travel along the tracks.

The position-time graph below shows the carts’ motion in a coordinate system. The graph shows straight, rising lines during the motion. This means that the position changes uniformly. In other words, it changes by the same magnitude every second. Looking at the graph, we can see that the red cart (cart 1) had a velocity of 0.31 m/s, meaning that it moved 0.31 m every second.

The blue cart had a higher velocity than the red one. This is indicated by the steeper line, which shows that the blue cart’s position increases faster. Velocity equals the change of position divided by the elapsed time, so the velocity is the position graph’s slope. The faster blue cart has a steeper slope than the red cart.

$v={\displaystyle \frac{\mathrm{\Delta <!--\; \Delta \; -->}x}{\mathrm{\Delta <!--\; \Delta \; -->}t}}$

In everyday life, one second is a short time and one metre is a short distance for certain situations involving motion. In traffic, distance is measured in kilometres and time in hours. The derived unit for velocity is km/h. The units can be converted as follows: