Particle Motion (by Ken Schwartz)
Interpreting one-dimensional motion on a two-dimensional graph can be confusing. Let's try to connect a graph with the actual movement of a particle.
When considering rectilinear (straight-line, one-dimensional) motion, we must always think of the moving object (often a sizeless, massless "particle") traveling along a straight line. It is either at rest (not moving), moving right, or moving left. (Up/down, backward/forward, and other opposite direction descriptors could be used as well). Yet when we look at the various motion graphs of position, velocity, and acceleration, what we see may not connect in our minds with the motion. Position refers to the location of the object relative to the origin. If it is located 5 units to the left of the origin, we say its position is . We often use or to represent the position function; that is, the function that gives the object's position as a function of time . (Geogebra only allows "" as a variable when defining functions, but think of it as "" in the app above). The blue graph shows the object's position along the -axis as a function of time. Notice that the blue graph is not a trace of the particle's path in the plane! If you click RESET and then START, you'll see that the particle moves along the -axis, its position at any time given by the -value of the point on the graph indicated with the cursor. Velocity is the particle's speed, with the direction of travel given by the sign of velocity. Velocity tells us how far a particle moves in a time period - that is, it tells us the rate of change of the particle's position. As such, velocity is the derivative of position: . When looking at the green velocity graph, you must connect the particle's speed, not its position, with the -value of the graph. Thus, the particle could be located above the origin, but its velocity could be negative. This only means that the particle's position is above but its movement is in the down direction. When the velocity is positive, the particle moves up; when it's negative, the particle moves down. The farther away from the -axis the -value is, the faster the particle is moving. Thus, speed is determined by how far away from the -axis the velocity is: . The object is not moving whenever (when the velocity graph meets the -axis), and the particle changes direction when changes sign (when the graph crosses the -axis). Acceleration is the particle's change in velocity over time - the rate of change of velocity. Therefore acceleration is the derivative of velocity , and so it is also the second derivative of position . Since we know that , imagine acceleration as the external force experienced by the particle. Acceleration is used in connection with velocity to determine when a particle is speeding up or slowing down. This is determined by comparing the signs of and : if they have the same sign, the particle is speeding up; otherwise it's slowing down. Graphically you can compare the sign of the red acceleration graph with the green velocity graph at any time .