# Graphical analysis of velocity and acceleration relationship

### a. the slope of a positiontime graph b. the

The area under an acceleration graph represents the change in velocity. In other words, the area under the acceleration graph for a certain time interval is equal. How to analyze graphs that relate velocity and time to acceleration and displacement. The slope of a velocity graph represents the acceleration of the object. Page 1. Graphical Analysis: Positions, Velocity and Acceleration (simple joints). ME Positional Analysis, The Starting Point; The Velocity Relationship.

Instantaneous velocity at any point is the slope of the tangent at that point.

Strategy The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point.

Determine the endpoints of the tangent. These correspond to a position of m at time 19 s and a position of m at time 32 s. Carrying this one step further, we note that the slope of a velocity versus time graph is acceleration. Acceleration versus time is graphed in Figure c. It is not accidental that the same equations are obtained by graphical analysis as by algebraic techniques.

Correlations imply physical relationships and might be shown by smooth graphs such as those above. From such graphs, mathematical relationships can sometimes be postulated.

## Graphs of Motion

This limit process is represented in the animation to the right. Seven tangents were added to our generic displacement-time graph in the animation shown above. Note that the slope is zero twice — once at the top of the bump at 3.

The slope of a horizontal line is zero, meaning that the object was motionless at those times.

### Graphical Analysis of Velocity and Acceleration

Since the graph is not flat, the object was only at rest for an instant before it began moving again. Although its position was not changing at that time, its velocity was. This is a notion that many people have difficulty with. It is possible to be accelerating and yet not be moving but only for an instant, of course.

### Graphs of Motion – The Physics Hypertextbook

Some interpret this as motion in reverse, but is this generally the case? Well, this is an abstract example. It's not accompanied by any text. Since the position of the bike increases by 4 m every second, the graph of x versus t is a straight line. Each point on this line gives the position of the bike at a particular time. In constructing the graph in Figure 2.

Suppose, however, that we were given this graph, but did not have prior knowledge of the velocity. The velocity could be determined by considering what happens to the bike between the times of 1 and 3 s, for instance. Notice that the slope is equal to the velocity of the bike.

Thus, for an object moving with a constant velocity, the slope of the straight line in a position-time graph gives the velocity.

Since the position-time graph is a straight line, any time interval Dt can be chosen to calculate the velocity.