 # Kinematics: Motion in One Dimension

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Kinematics: Motion in One Dimension

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

Essential Concepts Frames of reference Vector vs. scalar quantities
Displacement Velocity Average velocity Instantaneous velocity Acceleration Graphical representation of motion

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

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

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.

Displacement is change in position

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

Describing Motion Describing motion requires a frame of reference

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

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

Displacement Linear change in position of an object
Is not the same as distance

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

Reference Frames & Displacement
Direction is relative to the initial position, x1 x1 is the reference point

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

Average Velocity

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

Velocity can be represented graphically: Position Time Graphs

Velocity can be interpreted graphically: Position Time Graphs
Find the average velocity between t = 3 min to t = 8 min

Calculate the average velocity for the entire trip

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

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

Position-time when velocity is not constant

Average velocity compared to instantaneous velocity
Instantaneous velocity is the slope of the tangent line at any particular point in time.

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

2.2 Acceleration

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

X-t graph when velocity is changing

Acceleration Acceleration is the rate of change of velocity.

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.

Acceleration Because acceleration is a vector, it must have direction
Here is an example of negative acceleration:

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?

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

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

Velocity-Time Graphs Is this object accelerating? How do you know?
What can you say about its motion?

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?

Velocity-Time Graph dev.physicslab.org

Displacement with Constant Acceleration (C)

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

Displacement on v-t Graphs
Displacement is the area under the line!

Graphical Representation of Displacement during Constant Acceleration

Displacement on a Non-linear v-t graph
If displacement is the area under the v-t graph, how would you determine this area?

Final velocity of an accelerating object

Displacement During Constant Acceleration (D)

Graphical Representation

Derivation of the Equation

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.

Kinematic Equations

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.

Motion Graphs of Free Fall
v-t graph x-t graph

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!

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).

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

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?

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).

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

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.

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.

Displacement an v-t Curves
The displacement, x, is the area beneath the v vs. t curve.