Mechanics

# Graphic study of movements

Velocity x time (vxt) graph for uniform motion

In uniform motion (constant velocity), the graph representing velocity versus time (vxt) is a line parallel to the time axis.

Distance x time (dxt) graph for uniform motion

In any uniform motion, the distance x time graph is a line through the origin of the axes.

Slope of the graph

The slope of the dxt graph for a uniform motion gives us the value of the velocity of that motion, i.e.

What is the “position” of a mobile in its trajectory

To determine the position of a body on a trajectory, it is sufficient to provide the value of its distance, measured on the trajectory, to a point taken as a reference (origin). From this definition, suppose that a car is, at the instant t = 0, passing in front of the “kilometer 40” sign, as shown in the following figure. We say that the position of the car, relative to the beginning of the road, is s = 40 km (representing the position by the letter s). Obviously, this does not mean that the distance traveled by the car was at kilometer zero. Suppose, further, that the car continues on its journey and, at the instant t = 1 h, finds itself in front of the “kilometer 80” sign. So now the car’s position is s = 80 km (relative to the road). We can conclude that, in the time interval ∆d = 1 h (from t = 0 to t = 1 h),

Position x time graph

When a body moves along a given trajectory, with its velocity sometimes in one direction, sometimes in the opposite direction, it is customary to arbitrarily consider one of these speeds as positive and the other negative. In the case of the following example, the car on the road, the velocity of the movement “back” (in the decreasing direction of the kilometer marks) is usually considered negative. So, we could say that the speed of the car, in the last analyzed section, was v = – 80 km/h. As this convention is not useful in high school studies, teachers consider speed always positive.

Velocity x time (vxt) graph for motion with constant acceleration

The vxt graph for motion with constant acceleration is a line whose slope is equal to the value of the motion’s acceleration, that is,