On an \(x-t\) graph, the vertical axis represents position (\(\Delta x\)) and the horizontal axis represents time (\(\Delta t\)).
The rate of change of position:
\[ v = \text{slope} = \frac{\Delta x}{\Delta t} \]
Unit 1: Lesson 1 Slide 6 of 10
Guided Practice Calculating Velocity
2m 8m 0s 4s x (m) t (s)
1. Identify point 1: (0s, 2m)
2. Identify point 2: (4s, 8m)
3. Apply equation:
\( v = \frac{8 - 2}{4 - 0} = \frac{6}{4} = 1.5\text{ m/s} \)
Unit 1: Lesson 1 Slide 7 of 10
Concept Chunk 3 Curving Graphs
If the line on an \(x-t\) graph is curved, the slope is changing. This means the velocity is changing over time.
This indicates the presence of acceleration.
Key Vocabulary
Unit 1: Lesson 1 Slide 8 of 10
Classroom Challenge Translation Mastery
Look at the curved graph on the right. Discuss with your table group which motion diagram best models this profile:
x t
Is the velocity speeding up or slowing down?
Unit 1: Lesson 1 Slide 9 of 10
Unit Wrap Up Next Steps
You've mastered the fundamentals of position-time slopes and motion maps. Tomorrow, we explore Velocity-Time graphs, where slope represents acceleration and area represents displacement!
Complete the "Position Graphing Practice Worksheet" in your packet before tomorrow's class.
Unit 1: Lesson 1 Complete Slide 10 of 10
+v -v t
Explain how you would translate a parabolic curve on a position graph (speeding up) to a velocity graph. Fill in the blanks:
"Because the slope of the \(x-t\) curve is progressively over time, the physical velocity must be ."
"On the \(v-t\) graph, this changing velocity translates to a diagonal line with a non-zero , which physically represents constant ."
Unit 1: Kinematics — Translation Mastery Page 3 of 3
Pacing: 02:00
Unit 1: Lesson 2 Slide 5 of 10
Concept Chunk 2 Displacement Analysis
The mathematical product of the vertical unit (meters per second) and the horizontal unit (seconds) is meters.
\( \Delta x = \int v(t) \, dt \approx \text{Geometric Area} \)
Unit 1: Lesson 2 Slide 6 of 10
Guided Practice Constant Velocity Area
5 m/s 6 s Area = 5 x 6 = 30 m
When velocity is constant, the shape is a simple rectangle.
Displacement = Base x Height
Δx = (6 s) * (5 m/s) = +30 m
Unit 1: Lesson 2 Slide 7 of 10
Guided Practice Uniform Acceleration Area
10 m/s 4 s Area = ½ (4) (10)
When acceleration is uniform starting from rest, the shape is a triangle.
Displacement = ½ Base x Height
Δx = ½ (4 s) * (10 m/s) = +20 m
Unit 1: Lesson 2 Slide 8 of 10
Concept Chunk 3 Translating Graphs
To build a velocity-time graph from a position-time graph:
Interactive Translation Blueprint
Practice sketching this mapping on page 3 of your notes packet. Remember, segments with straight lines on position graphs yield step-like flat lines on velocity graphs!
Unit 1: Lesson 2 Slide 9 of 10
Unit Wrap Up Next Steps
You've completed Lesson 2! Tomorrow, we unlock the ultimate capstone challenge: The Kinematics Translator, where we combine equations, maps, and graphs into a single, cohesive motion-solving tool.
Complete the "Acceleration Analysis Worksheet" in your packet before tomorrow's class.
Unit 1: Lesson 2 Complete Slide 10 of 10
Swap packets with your partner. Review their drawings on page 2. Write constructive feedback below using this stem:
"Your velocity sketch is correct because its slope is ________, which matches the changing slope of their position graph. However, make sure that..."
Unit 1: Kinematics — Translation Writing Page 3 of 3
Pacing: 02:00
Unit 1: Lesson 3 Slide 5 of 10
Concept Chunk 2 The Capstone Matrix
To master translation, we solve a multi-tiered matrix. We analyze a single given representation and construct all other maps and sketches.
Open page 2 of your guide to track our guided example!
Unit 1: Lesson 3 Slide 6 of 10
Guided Practice Mapping the Parabola
x t
• Initial slope is zero \(\rightarrow\) \(v_0 = 0\).
• Slope becomes progressively steeper \(\rightarrow\) velocity increases.
• Therefore, the velocity graph is a straight line sloping upward from zero!
Unit 1: Lesson 3 Slide 7 of 10
Concept Chunk 3 AP Argumentation
To earn full credit on paragraph responses, your argument must be a coherent logical progression:
Claim \(\rightarrow\) Physical Rule \(\rightarrow\) Slope/Area connection \(\rightarrow\) Scientific Conclusion
Never write: "The line goes up."
Instead, write: "The slope of the position graph is positive and constant, which indicates a constant forward velocity."
Unit 1: Lesson 3 Slide 8 of 10
Class Discussion Free Fall Modeling
Think about throwing a ball straight up:
"What is the slope of the velocity-time graph for a falling object? Does it change when the object is at its maximum peak?"
Discuss as a class before opening the Capstone Assessment!
Unit 1: Lesson 3 Slide 9 of 10
Unit Completed Congratulations
You've completed the complete Kinematics Codebreakers series! You possess the coordinate, graphical, vector, and algebraic skills necessary to dismantle any 1D-kinematics problem on the AP Physics 1 exam.
Please clear your tables and prepare to complete the "Kinematics Translator Assessment".
Unit 1 Decoded Slide 10 of 10
Textbook Diagram 3.1: Slope as Velocity
Forward Constant Velocity (+v)
Slope = +v t x Value = constant t v
Backward Constant Velocity (-v)
Slope = -v t x Value = -v t v
Slope Definition:
On a position-time graph, slope is \(\frac{\Delta x}{\Delta t}\). Because velocity is displacement divided by time, slope has the physical meaning of velocity. Straight lines represent constant velocities, and flat lines represent at-rest intervals.
Unit 1: Kinematics Master Reader Page 3 of 13
Chapter 4
Page 4
When an object's speed magnitude increases over time, it is speeding up. On a motion map, this is characterized by progressive spacing expansion: the distance gaps between successive coordinate dots grow noticebly wider each second.
Simultaneously, the velocity vector arrows must grow longer at each subsequent dot to indicate the increasing speed. This scaling rule applies regardless of direction: arrows point forward and grow longer for positive speeding up, and point backward and grow longer for negative speeding up.
Textbook Diagram 4.1: Car Speeding Up (Growing Spacing)
Strobe Snapshots (t = 0s, 1s, 2s) Speeding Up: v and a point SAME direction
t = 0s t = 1s t = 2s
Stop & Jot
Look at the speeding up backward map. Explain why the arrows point left but grow longer. Does a larger backward arrow mean a higher speed?
Unit 1: Kinematics Master Reader Page 4 of 13
Chapter 5
Page 5
When an object's speed magnitude decreases, it is slowing down (or decelerating). On a motion map, this is characterized by shrinking spacing gaps: the distance covered in each consecutive second gets progressively smaller.
To represent this, we draw dots that get closer together over time, and velocity vector arrows that get progressively shorter at each subsequent coordinate point, eventually tapering to a stacked rest-dot if the object comes to a stop.
Textbook Diagram 5.1: Car Slowing Down (Shrinking Spacing)
Strobe Snapshots (t = 0s, 1s, 2s) Slowing Down: v and a point OPPOSITE directions
t = 0s t = 1s t = 2s
Think-Pair-Share
Compare the speeding up and slowing down maps. How can you look at dot spacing and velocity arrows to immediately tell if speed is decreasing?
Unit 1: Kinematics Master Reader Page 5 of 13
Chapter 6
Page 6
When velocity is changing, its graph profiles undergo major shape changes. On a position-time (\(x-t\)) graph, changing velocity is represented by a curved line (parabola).
The slope of the curve is constantly changing. To evaluate velocity at any instant, we draw a straight tangent line. If the curve gets steeper, speed is increasing. If it gets flatter, speed is decreasing. On a velocity-time (\(v-t\)) graph, changing velocity is represented by a sloped diagonal line.
Textbook Diagram 6.1: Curve-to-Slope Unification
Tangent Slopes get steeper: x-t curve gets steeper
Slope remains constant on v-t: v-t is straight diagonal
Unit 1: Kinematics Master Reader Page 6 of 13
Chapter 7
Page 7
Once we understand changing velocity, we define acceleration (\(a\)): a vector quantity measuring the rate of change of velocity over time.
Because velocity includes both magnitude (speed) and direction, an object experiences a non-zero acceleration if either of these properties changes. This yields exactly three physical ways to accelerate:
Textbook Diagram 7.1: The Three Ways to Accelerate
1. Speeding Up
|v| is increasing
2. Slowing Down
|v| is decreasing
3. Turning Direction
Direction vector changes
Stop & Jot
A car travels at a completely constant speed of \(25\text{ m/s}\) around a circular track. Is the car accelerating? Justify using the three ways to accelerate.
Unit 1: Kinematics Master Reader Page 7 of 13
Chapter 8
Page 8
Mathematically, acceleration measures how many meters-per-second of velocity are added or subtracted each elapsed second. Its metric unit is meters per second squared (\(\text{m/s}^2\)):
\[ a_{\text{avg}} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_0}{t_f - t_0} \]
While average acceleration measures coordinate change across a long timeframe segment, instantaneous acceleration measures the exact rate of change at a single precise second. In AP Physics 1, we focus on constant, uniform acceleration, where average and instantaneous accelerations are identical.
Textbook Box 8.2: Calculus Definition of Instantaneous Acceleration
"The instantaneous acceleration is the mathematical limit of the average acceleration as the time interval approaches zero:
\[ a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt} \]
On a velocity-time graph, this limit represents the exact slope of the tangent line at any singular instant."
Think-Pair-Share
A racecar accelerates from \(0\text{ m/s}\) to \(30\text{ m/s}\) in exactly \(5\text{ s}\). Calculate its average acceleration. Include units.
Unit 1: Kinematics Master Reader Page 8 of 13
Chapter 9
Page 9
An essential constant-acceleration scenario is free fall: vertical motion occurring solely under the influence of Earth's gravity, with air resistance neglected.
Regardless of the object's mass, Earth's gravity accelerates it downward at a constant rate, designated as \(g \approx 9.8\text{ m/s}^2\) (often approximated as \(10\text{ m/s}^2\)). Because this vector points downward, we write \(a = -g\).
Textbook Diagram 9.1: Trajectory Vector Profiles
v > 0 (Up) v = 0 (Peak) v < 0 (Down)
Key Trajectory Rules:
• Acceleration is strictly constant and downward (\(-10\text{ m/s}^2\)) at all points, including peak.
• Trajectory remains symmetric.
Stop & Jot
At the very peak of its vertical trajectory, a thrown ball has a velocity of zero. Is its acceleration also zero at this point? Explain.
Unit 1: Kinematics Master Reader Page 9 of 13
Chapter 10
Page 10
Because AP Physics 1 focuses on constant acceleration, our acceleration-time (\(a-t\)) graphs typically consist of horizontal lines. Bounded area measures change in velocity (\(\Delta v\)):
\[ \text{Area under } a\text{-}t = a \cdot \Delta t = \Delta v = v_f - v_0 \]
Textbook Diagram 10.1: Mapping Acceleration Area to Velocity Change
Area represents Δv: Area = +Δv t a
Corresponding Linear Increase on v-t: Slope = +a t v
Stop & Jot
If an object has an acceleration of zero, is it guaranteed to be stationary? Explain.
Unit 1: Kinematics Master Reader Page 10 of 13
Chapter 11
Page 11
The geometric area bounded between the velocity curve and the horizontal axis represents the object's displacement (\(\Delta x\)).
If the line forms a triangle, the displacement is \( \Delta x = \frac{1}{2} b h \). If it forms a rectangle, the displacement is \( \Delta x = b h \).
Textbook Diagram 11.1: Partitioning Trapezoidal Displacement Regions
Area 1: Rectangle (v₀·t) Area 2: Triangle (½ a·t²)
Total displacement proof:
Combining both regions yields the complete position formula:
\[ \Delta x = v_0 t + \frac{1}{2} a t^2 \]
Think-Pair-Share
What physical event does a negative displacement area (area located entirely below the time axis) represent? Explain.
Unit 1: Kinematics Master Reader Page 11 of 13
Chapter 12
Page 12
The ultimate test of mechanical fluency is mapping all representations together. Examine this full translation block of a constant accelerated motion starting from rest:
x vs. t Quadratic Curve
v vs. t Linear Diagonal
a vs. t Constant Level
Synthesizing Paragraph Argument
Complete these stems to build a coherent physical paragraph response:
"Because the tangent slope of the \(x-t\) graph becomes steadily , the corresponding velocity-time value must ."
"On the motion diagram, this acceleration translates to dots that are spaced ."
Think-Pair-Share peer reflection
Compare your completed booklet with a classmate. Work together to write one final question about free fall acceleration and solve it together below:
Unit 1: Kinematics Master Reader Complete Page 12 of 13
Chapter 13
Page 13
The kinematic "Big Three" formulas are directly linked to graphing slopes and areas under changing velocity profiles.
For example, the velocity slope formula \( a = \frac{v - v_0}{t} \) is rearranged to yield the first equation: \( v = v_0 + a t \). The position formula \( x = x_0 + v_0 t + \frac{1}{2} a t^2 \) represents the physical sum of the rectangular area (\(v_0 t\)) and the triangular area (\(\frac{1}{2} a t^2\)) under a velocity-time line!
Unifying Areas & Equations Rectangle Area = v₀·t Triangle Area = ½ a·t²
Final Checklist
Verify with a peer that you have mastered these core concepts:
Unit 1: Kinematics Master Reader COMPLETE Page 13 of 13
