Kinematics The Study of Motion Chapter 2

What are some different types of motion? What are some terms (concepts) that describe our observations of motion? Position What questions do they answer? Where? Distance How far? Time How long? When? Speed How fast? Acceleration How is the speed changing?

How do we operationally define the terms (concepts) that describe our observations of motion? Position Measured in terms of distances (coordinates) from the axes of a one, two, or three dimensional coordinate system.

Distance Change in position Measured as the total length of the line segments and/or curves that trace the path of the object’s motion. Units: meters, m Initial Position Final Position

Displacement The NET change in the position of an object. The length of the line segment joining the initial and final positions. Independent of the path followed. Units: meters, m

Speed The rate of change of position. Defined as the distance traveled divided by the time of travel. Note: Because speed can change during an object’s motion we will actually define average speed as the distance traveled divided by the time of travel. Speed may be: Zero... Constant but not zero... Changing... no motion object travels equal distances in equal time intervals object travels different distances in equal time intervals

Scalars and Vectors A scalar is any measured quantity having only a magnitude. A vector is any measured quantity having both a magnitude AND an associated direction. Examples: Mass Time Temperature Example: Displacement For motion in only one direction the direction can be specified with a + or - sign. Horizontal Motion:Toward the right (+)Toward the left (-) Vertical Motion: Up (+) Down (-) Speed Distance

+/- Signs in Various Contexts Used in front of a number: +20 Number is greater than zero -20 Number is less than zero Used in front of a vector: d = +20mDisplacement is 20m to the right (or up) d = -20m Displacement is 20m to the left (or down) Between two numbers or vectors: 20m+10mSum of two displacements20m+10m = 30m 20m-(-10m)Difference between two displacements 20m+(-10m) = 30m Indicates magnitude Indicates direction Indicates an operation

Vector Example: Displacement Horizontal Motion Vertical Motion

Other Vectors Speed is a scalar having only a magnitude. The corresponding vector is called velocity. velocity

Acceleration The rate of change in the velocity Acceleration is a vector. Acceleration may be: Zero… Constant but not zero... velocity constant. Velocity changing uniformly, by the same amount each unit of time. Changing... Velocity changing not uniformly, by the different amounts each unit of time. Note: In this course we will not consider motion with a changing acceleration.

Acceleration can change the velocity in three ways. If the acceleration is in the same direction as the velocity i.e., parallel to the velocity... The magnitude of the velocity increases i.e., the object moves faster. Pressing the gas pedal in a car does this. If the acceleration is in the opposite direction as the velocity i.e., antiparallel to the velocity... The magnitude of the velocity decreases i.e., the object moves slower. Pressing the brake pedal in a car does this. If the acceleration is perpendicular the velocity... The direction of the velocity changes. Turning the steering wheel of a car does this,

Consider an object projected upward. Because of gravity the object moves upward while at the same time going slower and slower. It momentarily stops at point “B”. It then moves downward while at the same time going faster and faster. From point “A” to point “B” : Displacement is + Velocity is + ; decreasing Acceleration is -, constant At point “B” : Velocity is zero Acceleration is -, constant From point “B” to point “C” : Displacement is - Velocity is - ; increasing Acceleration is -, constant

Summary of Concepts Concept Type Operational Definition Units Position Vector Measurements relative to varies 3-dimensional frame of reference. Distance Scalar Length of path from initial meters, m position to final position. Displacement Vector Length of line joining meters, m initial and final positions. (independent of path) Time Scalar Measured relative to some seconds, s periodic phenomena.

Summary of Concepts continued Concept Type Operational Definition Units Speed Scalar Rate of change of position. m/s Distance divided by time. Magnitude of velocity. Velocity Vector Rate of change of position. m/s Displacement divided by time. Acceleration Vector Rate of change of velocity. m/s 2 Change in velocity divided by time.

Symbolic-Mathematical Description of Motion Symbols:

Definitions:

We will investigate and describe two types of motion: Constant Velocity (acceleration = 0) Constant, nonzero, Acceleration Constant Velocity (acceleration = 0) Variables: displacement, d velocity, v…constant time, t Relationships: (from definition) a=0

Constant, nonzero, Acceleration Variables: displacement, d time, t Relationships: (from definition) Only if acceleration is constant

Other Derived Relationships for Uniformly Accelerated Motion Start with: Substitute: Simplify:

Start With: Substitute: Substitute Simplify:

Summary Constant Velocity Variables: dvtdvt Relationships: Constant Acceleration Variables:Relationships: a=0

Free-Fall Free-Fall includes all motion which meets the following two conditions: 1. Motion is only in the vertical direction (up/down) 2. Motion is only affected by gravity What type of motion is free-fall? Constant Velocity Constant Acceleration Does the acceleration depend on the mass of the object? Yes No What is the value of the acceleration due to gravity? Does it depend on the body producing the gravitational force?

Conclusion Free-Fall is an example uniformly accelerated motion. At the earth’s surface the acceleration due to gravity is constant and independent of the object’s mass. The acceleration due to gravity depends on the mass and size of the body producing the gravitational force. Does the acceleration depend on the direction (up/down)? Yes No