#AP Physics C: Mechanics - Motion in 1D - The Night Before 🚀

Hey! Let's get you prepped for tomorrow. We're going to break down motion in one dimension, focusing on the key stuff you absolutely need to know. Let's make this click!

#1. Representing Motion: Setting the Stage

Motion is all about how things move through space and time. We use different tools to describe it:

• Diagrams and Graphs: Visualizing motion 📊
• Equations: Describing motion mathematically
• Narratives: Explaining motion in words

#2. Kinematic Equations: Your Motion Toolbox

These equations are your best friends when dealing with constant acceleration. They allow us to predict an object's position and velocity at any time.

#The Big Three:

1. Velocity as a function of time: $v_{x}=v_{x 0}+a_{x} t$
   • This tells you the final velocity ($v_x$) after a time ($t$), given initial velocity ($v_{x0}$) and constant acceleration ($a_x$).
2. Position as a function of time: $x=x_{0}+v_{x 0} t+\frac{1}{2} a_{x} t^{2}$
   • Find the final position ($x$) based on initial position ($x_0$), initial velocity ($v_{x0}$), acceleration ($a_x$), and time ($t$).
3. Velocity as a function of position: $v_{x}^{2}=v_{x 0}^{2}+2 a_{x}\left(x-x_{0}\right)$
   • Relates final velocity ($v_x$) to initial velocity ($v_{x0}$), acceleration ($a_x$), and the change in position ($x - x_0$).

#3. Gravitational Acceleration: Falling Down 🌎

Near Earth's surface, gravity causes a constant downward acceleration:

• Direction: Downward
• Magnitude: Approximately $a_{g}=g \approx 10 \mathrm{~m} / \mathrm{s}^{2}$ (Use this on the exam!)

#4. Motion Graphs: Visualizing Change 📈

Graphs are powerful tools for understanding motion. Here's what you need to know:

#Position vs. Time (x vs. t):

• Slope: The slope of the tangent line at any point gives you the instantaneous velocity at that time. Mathematically, $v_{x}=\frac{d x}{d t}$

#Velocity vs. Time (v vs. t):

• Slope: The slope of the tangent line at any point gives you the instantaneous acceleration at that time. Mathematically, $a_{x}=\frac{d v_{x}}{d t}$
• Area under the curve: The area under the curve between two times gives you the displacement during that time interval. Mathematically, $\Delta x=\int_{t_{1}}^{t_{2}} v_{x}(t) d t$

#Acceleration vs. Time (a vs. t):

• Area under the curve: The area under the curve between two times gives you the change in velocity during that time interval. Mathematically, $\Delta v_{x}=\int_{t_{1}}^{t_{2}} a_{x}(t) d t$

Understanding how to extract information from these graphs is essential for both multiple-choice and free-response questions.

#Final Exam Focus 🎯

Okay, here’s the game plan for tomorrow:

• Kinematic Equations: Know them inside and out. Practice applying them in various scenarios.
• Motion Graphs: Be able to interpret them quickly. Practice finding slopes and areas under the curves.
• Free Fall: Understand that gravity provides constant acceleration. Practice solving free-fall problems.
• Time Management: Don't spend too long on one question. If you get stuck, move on and come back later.
• Units: Always include units in your answers.
• Show Your Work: Even if you make a mistake, you can get partial credit for showing your process.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. A ball is thrown vertically upward. At its highest point, which of the following is true? (A) Its velocity is zero, and its acceleration is zero. (B) Its velocity is zero, and its acceleration is nonzero. (C) Its velocity is nonzero, and its acceleration is zero. (D) Its velocity is nonzero, and its acceleration is nonzero.

2. A car accelerates uniformly from rest to a velocity of 20 m/s in 5 seconds. What is the car's acceleration? (A) 2 m/s² (B) 4 m/s² (C) 5 m/s² (D) 10 m/s²

3. The slope of a position vs. time graph represents: (A) Acceleration (B) Displacement (C) Velocity (D) Time

#Free Response Question

A small projectile is launched vertically upward from the ground with an initial velocity of $v_0 = 30 m/s$. Assume negligible air resistance and take $g = 10 m/s^2$.

(a) Calculate the maximum height reached by the projectile.

(b) Calculate the time it takes for the projectile to reach its maximum height.

(c) Calculate the total time the projectile is in the air before it returns to the ground.

(d) Sketch a velocity vs. time graph for the projectile's motion, indicating the times and velocities at key points in its trajectory.

#Scoring Guide:

(a) Maximum Height (3 points)

• 1 point for using the correct kinematic equation: $v_f^2 = v_0^2 + 2 a \Delta y$
• 1 point for setting $v_f = 0$ at the maximum height.
• 1 point for calculating the correct maximum height: $\Delta y = 45 m$

(b) Time to Reach Maximum Height (2 points)

• 1 point for using the correct kinematic equation: $v_f = v_0 + at$
• 1 point for calculating the correct time: $t = 3 s$

(c) Total Time in the Air (2 points)

• 1 point for recognizing that the time to go up equals the time to come down.
• 1 point for calculating the total time: $t_{total} = 6 s$

(d) Velocity vs. Time Graph (3 points)

• 1 point for a straight line with a negative slope.
• 1 point for the correct initial velocity (30 m/s).
• 1 point for the correct final velocity (-30 m/s) and time in the air (6 s).

You've got this! Go get 'em! 💪