A body moves with uniform rectilinear motion (u.r.m.) when it has constant velocity, i.e., when its trajectory is a straight line and its speed is constant. In this section, we are going to study the constant velocity motion graphs, also know as u.r.m. graphs, that is:

## U.R.M. Graphs

### Position-time (x-t) graph

$\overline{)x={x}_{0}+v\cdot t}$

The graph position-time (x-t) of a uniform rectilinear motion (u.r.m.). represents time on the horizontal axis (t-axis) and position on the vertical axis (x- axis). Observe as the position (normally the x-coordinate) increases (or decreases) uniformly with time. We can distinguish two cases, when the velocity is positive or when it is negative:

We can get the velocity from the angle α. To do it just remember that in a right triangle the tangent of each of its angles is defined as the opposite side (cathetus) divided by the adjacent one:

The value of the slope is the magnitude of the velocity. Therefore, the greater the slope of the straight line, the higher the velocity of the body.

### Velocity-time (v-t) graph

$\overline{)v={v}_{0}=\text{cnst}}$

The graph velocity-time (v-t) of a uniform rectilinear motion (u.r.m.) shows that the velocity remains constant over time. Again, we can distinguish two cases:

Notice that the area enclosed under the curve between two instants of time is the distance traveled.

Distance traveled graph

The area enclosed inside the straight line v-t, the abscissa axis and the times t and t0 corresponds to the distance traveled. This property is valid for any kind of motion.

In particular for the u.r.m., since it is the area is of a rectangle (base x height):

$∆x=x-{x}_{0}=v·\left(t-{t}_{0}\right)$

In this case, it is just the calculation of said area, since it is a rectangle. But, do you know what mathematical tool enables the calculation of the area under a curve, whatever its form?

### Acceleration-time (a-t) graph

$\overline{)a=0}$

The graph acceleration-time (a-t) of a uniform rectilinear motion (u.r.m.) shows that the acceleration is always zero. In this case, whether the velocity of the body is positive or negative, there is only one possibility, illustrated in the figure:

## Solved exercises worksheet

Here you can test what you have learned in this section.

#### Draw graphs from u.r.m. equations

difficulty

Determine the graphs of the following uniform rectilinear motions:

1. x = 3 + 4·t
2. x = 3 - 4·t
3. x = -3 + 4·t
4. x = -3 - 4·t
5. 3·x = 9 + 12·t

Where x is measured in meters and t in seconds.

#### x-t graph of a person walking

difficulty
A person walks in a straight line at a velocity of 5 km per hour for 15 minutes. Could you determine its graph of position with respect to time?

#### Analysis of uniform rectilinear motion graphs

difficulty

The following graph of position-time ( x-t ) corresponds to a body moving in a straight trajectory. Determine the equation of motion for each segment and its velocity-time graph. From the expression for each segment, search for a general expression in the form of a function defined by segments, called  a piecewise-defined function, for the position and one for the velocity.

## Formulas worksheet

Here is a full list of formulas for the section Constant Velocity Motion Graphs. By understanding each equation, you will be able to solve any problem that you may encounter at this level.

Click on the icon   to export them to any compatible external program.

#### Equation of position in rectilinear uniform motion -x-axis

$x={x}_{0}+v\cdot t$

#### Equation of acceleration in uniform rectilinear motion

$a=0$

#### Equation of velocity in uniform rectilinear motion

$v={v}_{0}=\text{cte}$