1. Motion

2. Kinematics  Kinematics is the study of motion.  Motion requires a change in position.  We graph position as a function of time.  Which point is farthest away?  Which point has the slowest motion?  Which point has the fastest motion?

3.  Position is a measure of the distance from a certain fixed point of reference.  The fixed point of reference is the origin equal to 0.  A position axis on a graph can give specific units, or it can give relative units using names of variables. Position Position (x) axis is vertical Units on this graph are relative

4. Displacement  Displacement measures the change in position compared to the starting point. Different from positionDifferent from position Measured relative to the object’s current positionMeasured relative to the object’s current position  In the previous example the axis shows position x 1 and x 2.  The displacement is  x = x 2 - x 1.

5. Speed and Velocity  Speed measures the rate of change of position along a path.  The direction doesn’t matter for speed – faster is faster in any direction. Forward or backwardsForward or backwards  Velocity measures the rate of change of position, and specifies the direction.  The rate of change of position is similar for speed and velocity: v =  x /  t.

6. Velocity in One Dimension  In one dimension objects only move on a straight line.  Objects can speed up and slow down.  Objects can go forward and backward.  Velocity is indicated with positive (forward) and negative (backwards) values. From P 1 to P 2 the position increases, and the velocity is positive. From P 3 to P 4 the position decreases, and the velocity is negative.

7. Average Velocity  The ratio v =  x /  t gives the average velocity during the time interval  t.  In the graph,  x = x 2 – x 1  x = x 2 – x 1  t = t 2 – t 1  t = t 2 – t 1  This is the slope of the line.

8. Short Times  If the time scale changes the velocity may also change.  The average velocity from P 1 to P 2 (green) is greater than the average velocity from P 1 to P A (orange). xAxA tAtA

9.  As t 2 gets closer to t 1,  t gets close to 0.  The above expression cannot be calculated at  t = 0.  The limit is needed. Limits  A mathematical limit describes the result of taking a value to a point or extreme. Extreme point  t = 0Extreme point  t = 0 No direct calculation of mathematical expressionNo direct calculation of mathematical expression  The limit of the velocity for short times is the instantaneous velocity. measures velocity at a pointmeasures velocity at a point

10. Position vs Time Graph  The instantaneous velocity appears as the slope of the graph at a point.  The tangent to the curve at a point is the slope at that point. This black line is tangent to the curve at P 1. The slope of that line is the instantaneous velocity at P 1.

11. The Derivative  The derivative measures the limit of the rate of change at each point.  The velocity is the derivative of the position. next v t