# 65 Graphing Motion

# Basic Motion Graphs

are a useful tool for visualizing and communicating information about an object’s motion. Our goal is to create motion graphs for our example skydiver, but first let’s make sure we get the basic idea.

We will start by looking at the motion graphs of on object with an of 2 **m** and of 4 **m/s**. An object moving at constant velocity has zero , so the graph of acceleration vs. time just remains at zero:

The velocity is constant, so the graph of velocity vs. time will remain at the 4 **m/s** value:

is the rate at which position changes, so the position v. time graph should change at a constant rate, starting from the (in our example, 2 **m**). The of a tells us the rate of change of the variable on the vertical axis, so we can understand as the slope of the position vs. time graph.

### Reinforcement Exercises

Now let’s look at for an object with . Let’s give our object the same of 2 **m**, and of 4 **m/s**, and now a constant acceleration of 2 **m/s/s**. The acceleration vs. time remains constant at 2 **m/s/s**:

is the rate at which changes, so acceleration is the of the velocity vs. time graph. For our constant 2 **m/s/s** acceleration the velocity graph should have a constant slope of 2 **m/s/s**:

Finally, if the is changing at a constant rate, then the of the position graph, which represents the velocity, must also be changing at a constant rate. The result of a changing slope is a curved graph, and specifically a curve with a constantly-changing slope is a *parabolic* curve, or a *parabola*.

We haven’t made for the situation of because they are relatively unexciting. The position graph is constant at the initial value of position, the graph is constant at zero and the graph is also constant at zero. Let’s end this section with some interesting graphs – those of an object that changes direction. For example, an object thrown into the air with an of 5 **m/s**, from an of 2 **m** that then falls to the ground at 0 **m**. Neglecting , the will be constant at negative *g, *or -9.8 **m/s/s**.

The velocity will be positive, but slowing down toward zero, cross through zero as the object turns around, and then begin increasing in the negative direction.

The position will increase as the object moves upward, then decrease as it falls back down, in a parabolic fashion because the slope is changing at a constant rate (acceleration is constant so velocity changes at a constant rate, so the slope of the position graph changes constantly).

Check out this interactive simulation of a moving person and the associated motion graphs:

### Reinforcement Exercises

### Everyday Example: Terminal Velocity

Let’s look at the for our skydiver while they are at a terminal velocity of -120 **MPH**, which is about 54 **m/s**. Let’s set our for this analysis to be the position where they hit terminal velocity.

Acceleration is zero because they are at terminal velocity:

Velocity is constant, but negative:

And position changes at a constant rate, becoming more negative with time.

### Everyday Example: Full Skydive

Now let’s look at the for our skydiver prior to reaching terminal velocity, starting from the initial jump.

graphs or plots with time on the horizontal axis and position or velocity or acceleration on the vertical axis

position at the start of the time interval over which motion is being analyzed

not changing, having the same value within a specified interval of time, space, or other physical variable

a quantity of speed with a defined direction, the change in speed per unit time, the slope of the position vs. time graph

the change in velocity per unit time, the slope of a velocity vs. time graph

the steepness of a line, defined as vertical change between two points (rise), divided by the horizontal change between the same two points (run)

the value of velocity at the start of the time interval over which motion is being analyzed

location in space defined relative to a chosen origin, or location where the value of position is zero

a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid