Velocity

Speed  Speed measures the rate of change of position along a path.  The direction doesn’t matter for speed, but path does. 9 m/s 12 m/s 3 m/s The start of the arrows all occur at the same time. The ends of the arrows all occur at the same time. Longer arrows mean faster motion. Not straight – not vectors.

Velocity  Velocity measures the rate of change of position, and specifies the direction.  The rate of change of position is similar for speed and velocity. 5 m/s northwest 8 m/s west 3 m/s northeast Velocity has speed and direction Velocity is a vector.

Velocity Magnitude  Displacement divided by time gives the average velocity.  The displacement vector divided by time gives the average velocity vector.  A person walks from 2 km north of the gym to 3 km west of the gym in 1.5 h. What is the magnitude of average velocity? The magnitude of displacement is The average velocity is the magnitude of displacement divided by the time.

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

 In one dimension objects only move on a straight line.  Objects can go forward and backward.  Objects can speed up and slow down. Motion in One Dimension Position (x) axis is vertical Units on this graph are relative 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.

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.

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 PAPA

 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 an extreme.  This is used when the result at the extreme point would give a mathematical expression that cannot be calculated.

Instananeous Velocity  The limit of the velocity for short times is the instantaneous velocity.  The instantaneous velocity appears as the slope of the graph at a point. This black line is tangent to the curve at P 1. The slope of that line is the instantaneous velocity at P 1.

Motion in Two Dimensions  Time is a scalar, not a vector. y x y x  Vector average velocity is parallel to the displacement.

Tangent  The average velocity becomes the instantaneous velocity for short time intervals.  The same is true for vectors.  The instantaneous velocity vector direction is tangent to the curve. y x next