Representations of Motion: AP Physics 1 Study Guide
Introduction
Buckle up, future physicists! Today we're diving into the world of motion, where objects dance to the tunes of displacement, velocity, and acceleration. Think of motion as the universe’s way of throwing a cosmic dance party, and we’re here to decipher every move. 🎉✨
Understanding Representations of Motion
In AP Physics 1, representing motion in various formats helps us understand and analyze how an object moves. Let’s break down these representations like Newton breaking down an apple (he saw gravity, we see graphs).
Graphical Representations of Motion 📈
Graphs are your best friends in Kinematics. They visually communicate the story of motion better than an Instagram reel.

PositionTime Graphs: These graphs plot how far an object travels over time. The slope of this graph gives you velocity. Fun fact: a straight line means constant velocity, whereas a curve means acceleration. Imagine your position at a superhero cinema marathon—your location on the couch changes linearly with time!

VelocityTime Graphs: These graphs illustrate an object’s velocity over time. The slope here tells you the acceleration, and the area under the curve reveals the displacement. Think of it like Santa's sleigh accelerating on a magic graph—more slope means more speed!

AccelerationTime Graphs: These show how acceleration changes over time. Not as common as the first two but equally thrilling. It's the physics equivalent of measuring how quickly Batman speeds up his Batmobile.
Numerical Representations
Sometimes we need raw numbers to tell the story of motion. Tables or lists showing position, velocity, and acceleration at different times are indispensable. Imagine Doctor Who’s TARDIS logs, documenting position and velocity at each point in time!
Analytical Representations
Mathematical equations are the language of the universe, beautifully describing motion. Equations involving variables like position (x), velocity (v), and acceleration (a) help predict future motion. For instance, the formula ( v_f = v_i + at ) is like predicting where your friend might end up next in a game of tag (if they never slow down).
Diagrammatic Representations
Sketches or diagrams visually represent an object’s position, velocity, and acceleration at different times. They’re like animated stick figures showing how far a sprinter has run, their speed, and how they accelerate towards a finish line.
Graphical Relationships 📊
The interconnectedness of these graphs is like the relationship web in your favorite drama series—each graph impacts the other.

From the slope of a PositionTime Graph, you find velocity. From the slope of a VelocityTime Graph, you get acceleration. Think of it like decoding codes in a mystery novel!

The area under the curve of a VelocityTime Graph gives you displacement, and for an AccelerationTime Graph, it provides velocity: it’s like unwrapping gifts to find what’s inside. 🎁
Center of Mass
Ah, the center of mass—the point where all the mass seems to concentrate. It's the balancing point, like where you’d balance a pencil on your finger.
 Formula: ( a = \frac{F}{m} )
 Key Insight: The acceleration of an object's center of mass is proportional to the net force acting on it.
Linearization 🤸♀️
Most natural phenomena are nonlinear. Physics says, "Let's cheat a bit and make it linear!"
By squarifying the xaxis values, curvy graphs become straight lines, making it easier to predict trends, like turning wavy spaghetti into straight rigid noodles. 🍝
Example
Imagine dropping your phone (sorry, but it helps).
 The equation: ( y = 10  4.9t^2 )
 To find where it’s at 1 second: ( y = 5.1 )
 Tangent line equation: ( y  5.1 = 0.49(x  1) )
This makes linearization your goto fixerupper for gnarly equations!
Mathematical Representations
Time to flex those algebra muscles with The Big Four equations of Kinematics:
 Horizontal displacement ( \Delta x ), Final and initial velocities ( v_f, v_i ), Time ( t ), and Acceleration ( a )
 Choose your formula like choosing a cookie from a jar: based on what's missing.
Example: Super Car
A supercar zooms at 68 m/s, decelerating at 4 m/s². What distance to stop?
 Equation: ( v_f^2 = v_i^2 + 2a\Delta x )
 Calculation: 0 = 4624  8(\Delta x)
 Result: Runway needed is 578 m.
Free Fall ⚽️
When gravity is the DJ, things fall freely.
 Key Vocab: Free Fall: Only force acting is gravity. Acceleration due to gravity (g): 9.8 m/s² (round to 10 if you’re feeling generous).
 Equations: Replace horizontal displacement ( \Delta x ) with vertical displacement ( \Delta y ).
Example: Dropped Ball
Drop a ball from a building. It falls for 2.8s.
 Displacement Equation: ( \Delta y = 0 + \frac{1}{2}(10)(2.8)^2 )
 Displacement: 39.2 meters.
 Velocity Equation: ( v_f = 0 + 10 \times 2.8 )
 Velocity: 28 m/s.
Projectile Motion ☄️
Let’s catapult into projectile motion, where objects follow a path under the influence of gravity alone—a parabolic trajectory, like the arc of Mario’s jump.
Key points:
 Horizontal Motion: Constant speed.
 Vertical Motion: Accelerated by gravity at 9.8 m/s².
 Trajectory: Parabolic path.
 Initial Velocity Components: Horizontal and vertical.
Example: Tennis Ball Off Ledge
Rolling off a 1.5m high ledge at 5 m/s.
 Time to hit ground: ( y = \frac{1}{2}gt^2 \Rightarrow 1.5 = 5t^2 \Rightarrow t = 0.55s )
 Horizontal Distance: ( \Delta x = v_xt = 5 \times 0.55 \Rightarrow 2.75m )
 Velocity upon impact: Vertical: ( v_y = gt = 5.5m/s )
Angled Motion 🏹
Discovering motion at an angle—where vectors split into horizontal (Vox) and vertical (Voy) components. It’s like breaking a pizza into slices.
 Vocabulary: Use cos for horizontal component, and sin for vertical.
 ( t = 2 \frac{Voy}{g} ) if start and end heights match.
Example: Cannonball at 30°
Shot at 20 m/s at a 30° angle.
 Time of flight: ( t = 2 \frac{20 \sin(30)}{10} )
 Range: ( x = 17.3 \times 2 = 34.6m )
 Max Height: ( y = \frac{(10)^2}{2 \times 10} = 5m )
Key Vocabulary and Concepts
 Acceleration: Rate of velocity change over time.
 Trajectory: Path of object under gravity.
 Vector Components: Break velocity into horizontal and vertical.
Conclusion
Congratulations, you’ve journeyed through the epic realms of motion! You now have the magical powers of interpreting graphs, solving equations, and predicting the whereabouts of flying objects like a physics wizard. Go forth, understand the cosmos, and ace that AP Physics 1 exam with the confidence of a superhero in a cape! 🦸♀️🎓