Mathematical Model of Motion Chapter 5. Velocity Equations Average velocity: v = d/ t To find the distance traveled with constant or average velocity.

Mathematical Model of Motion

Chapter 5

Velocity Equations

Average velocity: v = d/t To find the distance traveled with constant

or average velocity : d = vt To find an object’s location along the x

axis: x = x0 + vt

Constant Acceleration Equations

Final velocity after acceleration: v = v0 + at

Final position after acceleration d = ½ (v + v0)t

d = v0t + ½at2

Final velocity after acceleration v2 = v0

2 + 2a(d)

Free FallFree Fall

Objects accelerate as they fall Acceleration due to gravity is

9.80 meters per second per second

g = 9.80 m/s2

So speed of falling object increases almost 10 m/s every second it falls

Free Fall CalculationsFree Fall Calculations Instantaneous speed of

falling object = gt acceleration times the elapsed time of fall

To find distance of fall, d = 1/2 gt2

In any time interval, distance fallen = avg. speed times elapsed time

Terminal Velocity

Air resistance slows objects, depending on surface area and air density

Terminal velocity (speed) is reached when force of air resistance equals force due to gravity

In vacuum, all objects fall at same rate

Graphing Motion: position vs. time

Plot position (dependent variable) versus time (independent variable)

Sometimes d is used to represent position In horizontal motion, x is often used

instead To find position at any time, read the

values off the graph using the line

Graphing Motion: Position Graph with 2 or more objects

Motion for two or more objects can be plotted on the same graph

Where lines cross, objects are at the same position

Graphing Motion: Position Graph with Uniform Motion

Equal displacements in equal time intervals

Produces linear graph Slope of position graph gives velocity Horizontal line means zero velocity Upward slope means positive velocity Downward slope means negative velocity

Position Graph with Uniform Motion

Motion diagram to graph

Slope of a Graph

Graphing Motion: Position Graph with Acceleration

Graph will be curved Constant acceleration gives parabola Slope of graph at any point gives

instantaneous velocity

Position Graph with Acceleration

Converting motion diagram to graph

Velocity vs Time Graphs

Plot velocity as dependent variable, time as independent variable

Constant velocity gives horizontal line Slope of graph gives acceleration Positive acceleration gives upward slope Negative acceleration slopes downward If graph is linear, acceleration is constant

Velocity vs Time: acceleration

Comparing Uniform and Accelerated Motion Graphs

Uniform motion Accelerated Motion

Comparing Positive and Negative Velocity

Speeding up and Slowing Down

Velocity vs Time Graphs: Finding Displacement

Displacement can be found from velocity graph by finding the area between the graph and the time axis

Divide the area bounded by the graph line, the horizontal axis and the initial and final times into geometric sections (squares, rectangles, triangles) and find the area

Area below the time axis is negative displacement

Area under the Velocity Graph

Area Under the Velocity Graph

Divide complex areas into triangles and rectangles