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**Kinematics: Motion in One Dimension**

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**2.1 Displacement & Velocity Learning Objectives**

Describe motion in terms of displacement, time, and velocity Calculate the displacement of an object traveling at a known velocity for a specific time interval Construct and interpret graphs of position versus time

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**Essential Concepts Frames of reference Vector vs. scalar quantities**

Displacement Velocity Average velocity Instantaneous velocity Acceleration Graphical representation of motion

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**Reference Frames Motion is relative**

When we say an object is moving, we mean it is moving relative to something else (reference frame)

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**Scalar Quantities & Vector Quantities**

Scalar quantities have magnitude Example: speed 15 m/s Vector quantities have magnitude and direction Example: velocity 15 m/s North

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**Displacement ∆x = xf - xi Displacement is a vector quantity**

Indicates change in location (position) of a body ∆x = xf - xi It is specified by a magnitude and a direction. Is independent of the path traveled by an object.

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**Displacement is change in position**

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**Displacement vs. Distance**

Distance is the length of the path that an object travels Displacement is the change in position of an object

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**Describing Motion Describing motion requires a frame of reference**

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**Determining Displacement**

In these examples, position is determined with respect to the origin, displacement wrt x1

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**Indicating Direction of Displacement**

Direction can be indicated by sign, degrees, or geographical directions. When sign is used, it follows the conventions of a standard graph Positive Right Up Negative Left Down

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**Displacement Linear change in position of an object**

Is not the same as distance

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**Displacement Distance = length (blue)**

How many units did the object move? Displacement = change in position (red) How could you calculate the magnitude of line AB? ≈ 5.1 units, NE

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**Reference Frames & Displacement**

Direction is relative to the initial position, x1 x1 is the reference point

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Average Velocity Speed: how far an object travels in a given time interval Velocity includes directional information:

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Average Velocity

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Velocity Example A squirrel runs in a straight line, westerly direction from one tree to another, covering 55 meters in 32 seconds. Calculate the squirrel’s average velocity vavg = ∆x / ∆t vavg = 55 m / 32 s vavg = 1.7 m/s west

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**Velocity can be represented graphically: Position Time Graphs**

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**Velocity can be interpreted graphically: Position Time Graphs**

Find the average velocity between t = 3 min to t = 8 min

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**Calculate the average velocity for the entire trip**

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**Formative Assessment: Position-Time Graphs**

Object at rest? Traveling slowly in a positive direction? Traveling in a negative direction? Traveling quickly in a positive direction? dev.physicslab.org

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**Average vs. Instantaneous Velocity**

Velocity at any given moment in time or at a specific point in the object’s path

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**Position-time when velocity is not constant**

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**Average velocity compared to instantaneous velocity**

Instantaneous velocity is the slope of the tangent line at any particular point in time.

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**Instantaneous Velocity**

The instantaneous velocity is the average velocity, in the limit as the time interval becomes infinitesimally short.

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2.2 Acceleration

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**2.2 Acceleration Learning Objectives**

Describe motion in terms of changing velocity Compare graphical representations of accelerated and non-accelerated motions Apply kinematic equations to calculate distance, time, or velocity under conditions of constant acceleration

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**X-t graph when velocity is changing**

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Acceleration Acceleration is the rate of change of velocity.

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**Acceleration: Change in Velocity**

Acceleration is the rate of change of velocity a = ∆v/∆t a = (vf – vi) / (tf – ti) Since velocity is a vector quantity, velocity can change in magnitude or direction Acceleration occurs whenever there is a change in magnitude or direction of movement.

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**Acceleration Because acceleration is a vector, it must have direction**

Here is an example of negative acceleration:

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**Customary Dimensions of Acceleration**

a = ∆v/∆t = m/s/s = m/s2 Sample problems 2B A bus traveling at 9.0 m/s slows down with an average acceleration of -1.8 m/s. How long does it take to come to a stop?

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**Negative Acceleration**

Both velocity & acceleration can have (+) and (-) values Negative acceleration does not always mean an object is slowing down

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**Is an object speeding up or slowing down?**

Depends upon the signs of both velocity and acceleration Construct statement summarizing this table. Velocity Accel Motion + Speeding up in + dir - Speeding up in - dir Slowing down in + dir Slowing down in - dir

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**Velocity-Time Graphs Is this object accelerating? How do you know?**

What can you say about its motion?

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**Velocity-Time Graph Is this object accelerating? How do you know?**

What can you say about its motion? What feature of the graph represents acceleration?

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Velocity-Time Graph dev.physicslab.org

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**Displacement with Constant Acceleration (C)**

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**Displacement on v-t Graphs**

How can you find displacement on the v-t graph?

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**Displacement on v-t Graphs**

Displacement is the area under the line!

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**Graphical Representation of Displacement during Constant Acceleration**

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**Displacement on a Non-linear v-t graph**

If displacement is the area under the v-t graph, how would you determine this area?

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**Final velocity of an accelerating object**

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**Displacement During Constant Acceleration (D)**

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**Graphical Representation**

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**Derivation of the Equation**

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**Final velocity after any displacement (E)**

A baby sitter pushes a stroller from rest, accelerating at m/s2. Find the velocity after the stroller travels 4.75m. (p. 57) Identify the variables. Solve for the unknown. Substitute and solve.

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Kinematic Equations

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**2.3 Falling Objects Objectives**

Relate the motion of a freely falling body to motion with constant acceleration. Calculate displacement, velocity, and time at various points in the motion of a freely falling object. Compare the motions of different objects in free fall.

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**Motion Graphs of Free Fall**

v-t graph x-t graph

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Free Fall In the absence of air resistance, all objects fall to earth with a constant acceleration The rate of fall is independent of mass In a vacuum, heavy objects and light objects fall at the same rate. The acceleration of a free-falling object is the acceleration of gravity, g g = 9.81m/s2 memorize this value!

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Free Fall Free fall is the motion of a body when only the force due to gravity is acting on the body. The acceleration on an object in free fall is called the acceleration due to gravity, or free-fall acceleration. Free-fall acceleration is denoted with the symbols ag (generally) or g (on Earth’s surface).

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**Free Fall Acceleration**

Free-fall acceleration is the same for all objects, regardless of mass. This book will use the value g = 9.81 m/s2. Free-fall acceleration on Earth’s surface is –9.81 m/s2 at all points in the object’s motion. Consider a ball thrown up into the air. Moving upward: velocity is decreasing, acceleration is –9.81 m/s2 Top of path: velocity is zero, acceleration is –9.81 m/s2 Moving downward: velocity is increasing, acceleration is –9.81 m/s2

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**Sample Problem Falling Object**

A player hits a volleyball so that it moves with an initial velocity of 6.0 m/s straight upward. If the volleyball starts from 2.0 m above the floor, how long will it be in the air before it strikes the floor?

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**Sample Problem, continued**

1. Define Given: Unknown: vi = +6.0 m/s Δt = ? a = –g = –9.81 m/s2 Δ y = –2.0 m Diagram: Place the origin at the Starting point of the ball (yi = 0 at ti = 0).

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**2. Plan Choose an equation or situation: Both ∆t and vf are unknown.**

We can determine ∆t if we know vf Solve for vf then substitute & solve for ∆t 3. Calculate Rearrange the equation to isolate the unknowns: vf = m/s Δt = 1.50 s

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**Summary of Graphical Analysis of Linear Motion**

This is a graph of x vs. t for an object moving with constant velocity. The velocity is the slope of the x-t curve.

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**Comparison of v-t and x-t Curves**

On the left we have a graph of velocity vs. time for an object with varying velocity; on the right we have the resulting x vs. t curve. The instantaneous velocity is tangent to the curve at each point.

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**Displacement an v-t Curves**

The displacement, x, is the area beneath the v vs. t curve.

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