Describing Motion

the study of motion motion is a change in position two branches Mechanics

describes how objects move Kinematics Dynamics explains the causes of motion

Chapter 3 is about one- dimensional motion, as on a number line Mathematical Representations of Motion—The Basics

Origin: A reference point to the left—negative to the right—positive Mathematical Representations of Motion—The Basics

When motion is vertical: up—positive down—negative Mathematical Representations of Motion—The Basics

An object has moved if at one time its position is x 1 and at another time its position is x 2. Mathematical Representations of Motion—The Basics

An object’s position at a time can be represented by an ordered pair: (t 1, x 1 ) or (t 2, x 2 ) Mathematical Representations of Motion—The Basics

Displacement the change in position between two distinct points often different from the distance traveled

Scalars and Vectors a scalar contains just one piece of information a vector contains two: magnitude and direction vectors are represented in bold: d, v, etc.

Scalars and Vectors for vectors in one- dimensional motion, subscripts may be used, such as d x this will represent a change in position

What do we know about the family’s travels? a. displacement = 2 km north Since displacement is a vector, a direction must be indicated. Example 3-1

What do we know about the family’s travels? b. the car has traveled 10 km Since distance is a scalar, no direction needs to be indicated. Example 3-1

What do we know about the family’s travels? c. the displacement is zero, since its final and initial positions are the same When d = 0, no direction is necessary. Example 3-1

What do we know about the family’s travels? d. the car has traveled 20 km Example 3-1

plots ordered pairs of data in a simple form Position-time Graph

allows the calculation of: displacement average speed Position-time Graph

to calculate: Average Speed v = |x 2 - x 1 | t 2 - t 1 = s ΔtΔt = |Δx| ΔtΔt

the speed of an object at any one moment the slope of the position- time curve at that point Instantaneous Speed

The slope is easy to find if the position-time curve is linear, but what if it is a curve? We can use a tangent line. Instantaneous Speed

Can you see why graph (c) is the best estimate for a tangent line? Instantaneous Speed

Be sure to recognize the difference between average speed and instantaneous speed. For which one can you get a speeding ticket?? Instantaneous Speed

Velocity includes both speed and direction to calculate average velocity: v = d ΔtΔt

Velocity displacement ( d ) might be positive or negative in one-dimensional motion v = d ΔtΔt

Velocity can be calculated from a position-time graph can be positive or negative v = d ΔtΔt

allows the calculation of: acceleration Velocity-time Graph

Acceleration change in velocity with respect to time to calculate average acceleration: a = ΔvΔv ΔtΔt

Acceleration acceleration is a vector pointing in the same direction as Δ v a = ΔvΔv ΔtΔt

Acceleration average acceleration can be calculated as the slope of a velocity- time graph a = ΔvΔv ΔtΔt

Acceleration uniformly accelerated motion involves a constant rate of velocity change

Equations of Motion

First Equation of Motion often used if you want to know the final velocity when you know the initial velocity and acceleration v 2x = v 1x + a x Δt

Determining Displacement Algebraically d x = ½(v 1x + v 2x )Δt d x = v x Δt

Determining Displacement Geometrically the area “under the curve” of a velocity-time graph is equal to the displacement of the moving object

Second Equation of Motion two common forms: d x = v 1x Δt + ½a x (Δt)² x 2 = x 1 + v 1x Δt + ½a x (Δt)²

Third Equation of Motion two common forms: d x = v 2x ² – v 1x ² 2a x x 2 = x 1 + v 2x ² – v 1x ² 2a x

Equations of Motion These are used to solve most problems involving straight-line, constant acceleration motion. Sometimes there will be more than one possible method.

Free Fall an object falls under the influence of gravity alone with negligible air resistance near earth’s surface: g = g y = m/s²

Free Fall the equations of motion are easily adapted by replacing the acceleration with g y : v 2y = v 1y + g y Δt First Equation of Motion:

Free Fall Second Equation of Motion: d y = v 1y Δt + ½g y (Δt)² Third Equation of Motion: 2g y d y = v 2y ² – v 1y ²