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**Kinematics in One Dimension**

Chapter 2 Kinematics in One Dimension

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**Kinematics deals with the concepts that**

are needed to describe motion. Dynamics deals with the effect that forces have on motion. Together, kinematics and dynamics form the branch of physics known as Mechanics.

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2.1 Displacement

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2.1 Displacement

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2.1 Displacement

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2.1 Displacement

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**SI units for speed: meters per second (m/s)**

2.2 Speed and Velocity Average speed is the distance traveled divided by the time required to cover the distance. SI units for speed: meters per second (m/s)

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**Example 1 Distance Run by a Jogger**

2.2 Speed and Velocity Example 1 Distance Run by a Jogger How far does a jogger run in 1.5 hours (5400 s) if his average speed is 2.22 m/s?

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**Average velocity is the displacement divided by the elapsed time.**

2.2 Speed and Velocity Average velocity is the displacement divided by the elapsed time.

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2.2 Speed and Velocity Example 2 The World’s Fastest Jet-Engine Car Andy Green in the car ThrustSSC set a world record of 341.1 m/s in To establish such a record, the driver makes two runs through the course, one in each direction, to nullify wind effects. From the data, determine the average velocity for each run.

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2.2 Speed and Velocity

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**The notion of acceleration emerges when a change in **

velocity is combined with the time during which the change occurs.

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**DEFINITION OF AVERAGE ACCELERATION**

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2.3 Acceleration Example 3 Acceleration and Increasing Velocity Determine the average acceleration of the plane.

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2.3 Acceleration

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2.3 Acceleration Example 3 Acceleration and Decreasing Velocity

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**2.4 Equations of Kinematics for Constant Acceleration**

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**2.4 Equations of Kinematics for Constant Acceleration**

Five kinematic variables: 1. displacement, x 2. acceleration (constant), a 3. final velocity (at time t), v 4. initial velocity, vo 5. elapsed time, t

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**2.4 Equations of Kinematics for Constant Acceleration**

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**2.4 Equations of Kinematics for Constant Acceleration**

Example 6 Catapulting a Jet Find its displacement.

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**2.4 Equations of Kinematics for Constant Acceleration**

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**2.5 Applications of the Equations of Kinematics**

Reasoning Strategy 1. Make a drawing. 2. Decide which directions are to be called positive (+) and negative (-). 3. Write down the values that are given for any of the five kinematic variables. 4. Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. 5. When the motion is divided into segments, remember that the final velocity of one segment is the initial velocity for the next. 6. Keep in mind that there may be two possible answers to a kinematics problem.

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**2.5 Applications of the Equations of Kinematics**

Example 8 An Accelerating Spacecraft A spacecraft is traveling with a velocity of m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing? x a v vo t m -10.0 m/s2 ? +3250 m/s

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**2.5 Applications of the Equations of Kinematics**

x a v vo t m -10.0 m/s2 ? +3250 m/s

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**In the absence of air resistance, it is found that all bodies **

2.6 Freely Falling Bodies In the absence of air resistance, it is found that all bodies at the same location above the Earth fall vertically with the same acceleration. This idealized motion is called free-fall and the acceleration of a freely falling body is called the acceleration due to gravity.

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2.6 Freely Falling Bodies

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**Example 10 A Falling Stone **

2.6 Freely Falling Bodies Example 10 A Falling Stone A stone is dropped from the top of a tall building. After 3.00s of free fall, what is the displacement y of the stone?

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2.6 Freely Falling Bodies y a v vo t ? -9.80 m/s2 0 m/s 3.00 s

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2.6 Freely Falling Bodies y a v vo t ? -9.80 m/s2 0 m/s 3.00 s

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**Example 12 How High Does it Go? The referee tosses the coin up **

2.6 Freely Falling Bodies Example 12 How High Does it Go? The referee tosses the coin up with an initial speed of 5.00m/s. In the absence if air resistance, how high does the coin go above its point of release?

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2.6 Freely Falling Bodies y a v vo t ? -9.80 m/s2 0 m/s +5.00 m/s

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2.6 Freely Falling Bodies y a v vo t ? -9.80 m/s2 0 m/s +5.00 m/s

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**Conceptual Example 14 Acceleration Versus Velocity **

2.6 Freely Falling Bodies Conceptual Example 14 Acceleration Versus Velocity There are three parts to the motion of the coin. On the way up, the coin has a vector velocity that is directed upward and has decreasing magnitude. At the top of its path, the coin momentarily has zero velocity. On the way down, the coin has downward-pointing velocity with an increasing magnitude. In the absence of air resistance, does the acceleration of the coin, like the velocity, change from one part to another?

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**Conceptual Example 15 Taking Advantage of Symmetry **

2.6 Freely Falling Bodies Conceptual Example 15 Taking Advantage of Symmetry Does the pellet in part b strike the ground beneath the cliff with a smaller, greater, or the same speed as the pellet in part a?

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Position-Time Graphs We can use a position-time graph to illustrate the motion of an object. Position is on the y-axis Time is on the x-axis

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**Plotting a Distance-Time Graph**

Axis Distance (position) on y-axis (vertical) Time on x-axis (horizontal) Slope is the velocity Steeper slope = faster No slope (horizontal line) = staying still

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Where and When We can use a position time graph to tell us where an object is at any moment in time. Where was the car at 4 s? 30 m How long did it take the car to travel 20 m? 3.2 s

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Interpret this graph…

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Describing in Words

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**Describing in Words Describe the motion of the object.**

When is the object moving in the positive direction? Negative direction. When is the object stopped? When is the object moving the fastest? The slowest?

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Accelerated Motion In a position/displacement time graph a straight line denotes constant velocity. In a position/displacement time graph a curved line denotes changing velocity (acceleration). The instantaneous velocity is a line tangent to the curve.

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Accelerated Motion In a velocity time graph a line with no slope means constant velocity and no acceleration. In a velocity time graph a sloping line means a changing velocity and the object is accelerating.

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**Velocity Velocity changes when an object… Speeds Up Slows Down**

Change direction

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**Velocity-Time Graphs Velocity is placed on the vertical or y-axis.**

Time is place on the horizontal or x-axis. We can interpret the motion of an object using a velocity-time graph.

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Constant Velocity Objects with a constant velocity have no acceleration This is graphed as a flat line on a velocity time graph.

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Changing Velocity Objects with a changing velocity are undergoing acceleration. Acceleration is represented on a velocity time graph as a sloped line.

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**Positive and Negative Velocity**

The first set of graphs show an object traveling in a positive direction. The second set of graphs show an object traveling in a negative direction.

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**Speeding Up and Slowing Down**

The graphs on the left represent an object speeding up. The graphs on the right represent an object that is slowing down.

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Two Stage Rocket Between which time does the rocket have the greatest acceleration? At which point does the velocity of the rocket change.

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**Displacement from a Velocity-Time Graph**

The shaded region under a velocity time graph represents the displacement of the object. The method used to find the area under a line on a velocity-time graph depends on whether the section bounded by the line and the axes is a rectangle, a triangle

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**2.7 Graphical Analysis of Velocity and Acceleration**

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**2.7 Graphical Analysis of Velocity and Acceleration**

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**2.7 Graphical Analysis of Velocity and Acceleration**

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