1. Graphs of Motion Physics Notes

2. Introduction to Graphs  Graphs are mathematical pictures.  They are the best way to convey a description of real world events in a compact form.  Graphs of motion come in several types depending upon which of the kinematic quantities (time, displacement, velociy, or acceleration) are assigned to which axis.

3. Displacement/Time  Slope equals velocity.  The “y” intercept equals the initial displacement.  When two curves coincide, the two objects have the same displacement at that time.

4. Displacement/time

5.  Straight lines imply constant velocity.  Curved lines imply acceleration.  An object undergoing constant acceleration traces a portion of a parabola.  Average velocity is the slope of the straight line connecting the endpoints of a curve.

6. Displacement/time  When the slope is negative, the object is traveling in the negative direction.  Positive slope implies motion in the positive direction.  Negative slope implies motion in the negative direction.  Zero slope implies a state of rest.

7. Displacement/time

8. Velocity/time  Not the same as displacement/time graphs.  In a velocity/time graph, higher means faster not farther.  Slope equals acceleration.  The “y” intercept equals the initial velocity.  When two curves coincide, the two objects have the same velocity at that time.

9. Velocity/time  Straight lines imply uniform acceleration.  Curved lines imply non-uniform acceleration.  An object undergoing constant acceleration traces a straight line.

10. Velocity/time

11.  Since a curved line has no single slope, we must decide what we mean when we asked for the acceleration of an object.  If the average acceleration is desired, draw a line connecting the endpoints of the curve and calculate slope.  If the instantaneous acceleration is desired, take the limit of this slope as the time interval shrinks to zero, that means take the slope of a tangent to a curve at any point.

12. Velocity/time

13.  Positive slope implies an increase in velocity in the positive direction.  Negative slope implies an increase in velocity in the negative direction.  Zero slope implies motion with constant velocity.

14. How to determine displacement, velocity, and acceleration?  Given a graph of any of these quantities (displacement, velocity, or acceleration), it is always possible in principle to determine the other two.

15. How to determine displacement, velocity, and acceleration from a graph? When acceleration is constant, the average velocity is just the average of the initial and final values in an interval. To find the displacement of the object during a specific interval, simply find the area under each segment of the line. See example on next slide.

17. Fundamentals of physics  The first derivative of displacement with respect to time is velocity.  The derivative of a function is the slope of a line tangent to its curve at a given point.  The inverse operation of the derivative is called the integral

18. Fundamentals of physics  The integral of a function is the cumulative area between the curve and the horizontal axis over some interval.  This inverse relation between the actions of derivative (slope) and integral (area) it is called the fundamental theorem of calculus.

19. Acceleration/time  Acceleration/time graphs of objects that have a constant velocity will be a horizontal line collinear with a horizontal axis.  On this type of graph…slope is meaningless.  The “y” intercept equals the initial acceleration.

20. Acceleration/time  When two curves coincide, the two objects have the same acceleration at that time.  An object undergoing constant acceleration traces a horizontal line.  Zero slope implies motion with constant acceleration.  Acceleration is the rate of change of velocity with time.

21. Acceleration/time

22.  Transforming a velocity/time graph to acceleration/time graph means calculating the slope of a line tangent to the curve at any point (this is called finding the derivative).  The reverse process involves calculating the cumulative area under the curve (finding the integral).  The area under the curve equals the change in velocity.