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Transcript of Kinematics: Motion in One Dimension. 2.1 Displacement & Velocity Learning Objectives Describe motion...
![Page 1: Kinematics: Motion in One Dimension. 2.1 Displacement & Velocity Learning Objectives Describe motion in terms of displacement, time, and velocity Calculate.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649e0c5503460f94af5b7e/html5/thumbnails/1.jpg)
Kinematics:Motion in One Dimension
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2.1 Displacement & VelocityLearning 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
• 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
www.cnx.org
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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 MotionDescribing motion requires a frame of reference
http://www.sfu.ca/phys/100/lectures/lecture5/lecture5.html
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Determining DisplacementIn these examples, position is determined with respect to the origin, displacement wrt x1
http://www.sfu.ca/phys/100/lectures/lecture5/lecture5.html
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Indicating Direction of Displacement
When sign is used, it follows the conventions of a standard graph
Positive Right Up
Negative Left Down
Direction can be indicated by sign, degrees, or geographical directions.
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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
dev.physicslab.org
Object at rest?
Traveling slowly in a positive direction?
Traveling in a negative direction?
Traveling quickly in a positive direction?
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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 AccelerationLearning 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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AccelerationAcceleration 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 2BA 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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www.gcsescience.com
Velocity-Time Graphs• Is this object accelerating?• How do you know?• What can you say about its motion?
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www.gcsescience.com
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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dev.physicslab.org
Velocity-Time Graph
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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?
tvxt
xv
so ,
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Displacement on v-t Graphs
tvx 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)
xavv if 222
A baby sitter pushes a stroller from rest, accelerating at 0.500 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
xavvtatvx
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2 2
1
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222
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2.3 Falling Objects
Objectives
1. Relate the motion of a freely falling body to motion with constant acceleration.
2. Calculate displacement, velocity, and time at various points in the motion of a freely falling object.
3. Compare the motions of different objects in free fall.
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Motion Graphs of Free Fall
x-t graph v-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 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).
Free Fall
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• 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
Free Fall Acceleration
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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. DefineGiven: 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:
yavv if 222 tavv if
yavv if 22a
vvt if
vf = - 8.7 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.