#AP Physics 1: Motion Mastery - Your Ultimate Study Guide 🚀

Hey there, future physics pro! This guide is designed to be your go-to resource for acing the AP Physics 1 exam, especially when you're in the final stretch. Let's make sure you're confident and ready to tackle anything they throw at you! 🎯

#Representations of Motion 🚂

Understanding motion is all about how we represent it. Here's a quick rundown:

1. Graphical Representations: Graphs are your friends! They show relationships between position, velocity, and acceleration over time. Check out the Graphical Representations of Motion section for more details.
2. Numerical Representations: Tables and lists of data points help you see the specifics of motion at different moments. 📊
3. Analytical Representations: These are the mathematical equations that describe motion. They're super useful for making predictions. See Mathematical Representations of Motion for the key equations. 🧮
4. Diagrammatic Representations: Sketches and diagrams help you visualize what's happening. ✍️

#Center of Mass ⚖️

The center of mass is like the balance point of an object or system. It's the point where you can apply a force to cause linear acceleration without any rotation. Think of it as the average location of all the mass. 📍

#Essential Knowledge:

• 4.A.1: The linear motion of a system is described by the displacement, velocity, and acceleration of its center of mass.
• 4.A.2: Remember that acceleration is the rate of change of velocity, and velocity is the rate of change of position. ⏰

#Graphical Representations of Motion 📈

Graphs are a fantastic way to visualize motion. Here's how to interpret them:

#Position vs. Time Graphs

• The slope of a position-time graph gives you the velocity. 🏃
• The area under the curve isn't directly useful here, but the change in position (displacement) is.

#Velocity vs. Time Graphs

• The slope of a velocity-time graph gives you the acceleration. 🚀
• The area under the curve gives you the displacement. 📏

#Acceleration vs. Time Graphs

• The area under the curve gives you the change in velocity. 💨

#Image Courtesy of geogebra

#Linearization 🤸‍♀️

Sometimes, you'll encounter curved graphs. Don't panic! Linearization is a technique to turn those curves into straight lines by manipulating the x-axis. This makes it easier to analyze the data. For example, if you have a graph of position vs. time squared, you might get a straight line.

#Image Courtesy of x-engineer.org

#Linearization Key Points:

• A linear function is a straight line: $y = mx + b$.
• A nonlinear function is anything that isn't a straight line.
• We approximate a nonlinear function with a tangent line to make analysis easier.
• The tangent line equation is: $y - y_1 = m(x - x_1)$.

Example Problem

Let's say a ball is dropped from 10 meters, and we want to find the time it takes to hit the ground. The position of the ball as a function of time is:

$y = 10 - 4.9t^2$

This is nonlinear, so let's linearize it. At $t = 1$ second, $y = 5.1$ meters. To find the tangent line, we can use the point at $t=1$ and a nearby point like $t=1.1$. At $t=1.1$, $y=4.61$. The slope of the tangent line at $t=1$ is approximately $(4.61 - 5.1)/(1.1 - 1) = -4.9$.

The tangent line equation is then:

$y - 5.1 = -4.9(t - 1)$ or $y = -4.9t + 10$

Using this linear equation, we can approximate the position of the ball at different times.

#Mathematical Representations of Motion 📍

These are your go-to equations for solving kinematics problems. Make sure you know them inside and out! 🧠

Variable Interpretation:

• Δx = horizontal displacement (m)
• Vf = final velocity (m/s)
• Vo = initial velocity (m/s)
• t = time (s)
• a = acceleration (m/s²)

Example: A car moving at 68 m/s slows down at a rate of 4 m/s². How much runway is needed to stop? We know $a$, $V_o$, and $V_f$ (which is 0). We're missing $\Delta x$ and $t$. Use the equation:

$V_f^2 = V_o^2 + 2a\Delta x$

Plugging in values:

$0 = (68 m/s)^2 + 2(-4 m/s^2)(\Delta x)$

Solving for $\Delta x$, we get $\Delta x = 578$ m.

#Free Fall ⚽️

Free fall is when an object is only under the influence of gravity. The acceleration due to gravity is approximately $g = 9.8 m/s^2$ (or 10 m/s² for simplicity on the AP exam). 🌍

• Key Vocabulary: Free Fall - an object only under the influence of gravity.
• Equation: velocity = force of gravity x time
• Key Vocabulary: Acceleration due to Gravity - 9.8 m/s/s (it is acceptable to round up to 10 m/s/s on the AP Physics 1 exam)

Here are the modified 'Big Four' equations for free fall:

Variable Interpretation:

• Δy = vertical displacement (m)
• Vf = final velocity (m/s)
• Vo = initial velocity (m/s)
• t = time (s)
• g = acceleration due to gravity (m/s²)

Example #1: A ball is dropped from a building and falls for 2.8 seconds. What is its displacement? We know $g$, $t$, and $V_o$ (which is 0). We're missing $\Delta y$ and $V_f$. Use the equation:

$\Delta y = V_ot + \frac{1}{2} gt^2$

Plugging in values:

$\Delta y = \frac{1}{2} (10 m/s^2)(2.8 s)^2$

$\Delta y = 39.2$ m

What is the final velocity? Use the equation:

$V_f = V_o + gt$

$V_f = (10 m/s^2)(2.8 s)$

$V_f = 28$ m/s

Example #2: A soccer ball is thrown straight up with an initial velocity of 20 m/s. What height does it reach? We know $g$, $V_o$, and $V_f$ (which is 0 at max height). We're missing $\Delta y$ and $t$. Use the equation:

$V_f^2 = V_o^2 + 2g\Delta y$

$0 = (20 m/s)^2 + 2(-10 m/s^2)(\Delta y)$

$\Delta y = 20$ m

#Projectile Motion ☄️

Projectile motion is when an object is launched into the air and is only affected by gravity. Think of a ball thrown through the air. 🏀

Key points:

• Motion is in two dimensions: horizontal and vertical.
• Horizontal motion is at a constant speed (no acceleration).
• Vertical motion is affected by gravity (constant downward acceleration).
• The path is a parabola.
• Ignore air resistance.

When dealing with horizontal projectile launches, we break it up into two sets of equations: vertical and horizontal.

Variable Interpretation:

• Δy = vertical displacement (m)
• Δx = horizontal displacement (m)
• $V_{fy}$ = vertical final velocity (m/s)
• $V_{oy}$ = vertical initial velocity (m/s)
• $V_x$ = horizontal velocity (m/s)
• t = time (s)
• g = acceleration due to gravity (m/s²)

Example: A tennis ball rolls off a 1.5m high table with a velocity of 5 m/s. How long does it take to hit the ground, how far does it travel and what is the final velocity?

(a) Time to hit the ground:

$\Delta y = V_{oy}t + \frac{1}{2} gt^2$

Since $V_{oy} = 0$:

$1.5 m = \frac{1}{2} (10 m/s^2) t^2$

$t = 0.55$ s

(b) Horizontal distance:

$\Delta x = V_x t$

$\Delta x = (5 m/s)(0.55 s)$

$\Delta x = 2.75$ m

(c) Final velocity:

$V_{fy} = V_{oy} + gt$ or $V_{fy}^2 = V_{oy}^2 + 2g\Delta y$

$V_{fy} = (10 m/s^2)(0.55 s)$

$V_{fy} = 5.5$ m/s

#Angled Motion 🏹

Angled launches involve an initial velocity with both horizontal and vertical components. You'll need to break the initial velocity into its components. 📐

• Key Vocabulary: Angled Launches - launches at an angle that includes both a horizontal and vertical component of initial velocity

• Key Vocabulary: Vector Components - the horizontal and vertical parts of a vector

• If an object is going upward, $V_{oy}$ is (+).

• If an object is going downward, $V_{oy}$ is (-).

• Vertical velocity is 0 at the top.

• Flight is symmetric if the projectile starts and ends at the same height.

Variable Interpretation:

• $V_{oy}$ = vertical initial velocity (m/s)
• $V_{ox}$ = initial horizontal velocity (m/s)
• t = time (s)
• g = acceleration due to gravity (m/s²)

Example: A cannonball is shot at a 30-degree angle above the horizontal at 20 m/s.

(a) Time in the air:

$t = 2(V_{oy})/g$ and $V_{oy} = V_o\sin(\theta)$

$V_{oy} = 20\sin(30) = 10$ m/s

$t = 2(10 m/s)/(10 m/s^2) = 2$ s

(b) Horizontal distance:

$\Delta x = V_x t$ and $V_{ox} = V_o\cos(\theta)$

$V_{ox} = 20\cos(30) = 17.3$ m/s

$\Delta x = (17.3 m/s)(2 s) = 34.6$ m

(c) Maximum height:

$V_{fy}^2 = V_{oy}^2 + 2g\Delta y$

$0 = (10 m/s)^2 + 2(-10 m/s^2)\Delta y$

$\Delta y = 5$ m

#Final Exam Focus 🎯

Okay, you've made it this far! Here's what to focus on for the exam:

Kinematics Graphs: Be able to interpret position, velocity, and acceleration graphs. Pay special attention to slopes and areas! 📈

The Big Four Equations: Know how to use these for both linear and free-fall motion. 🧮

Projectile Motion: Understand how to separate horizontal and vertical components. ☄️

• Linearization: Be able to linearize graphs and understand why it's useful. 🤸‍♀️
• Center of Mass: Remember the basics and how it relates to linear motion. ⚖️

#Last-Minute Tips:

• Time Management: Don't spend too long on one question. If you're stuck, move on and come back later. ⏰
• Common Pitfalls: Watch out for mixing horizontal and vertical components in projectile motion. Double-check your units! ⚠️
• Challenging Questions: Break down complex problems into smaller, manageable steps. Draw diagrams and write down what you know. 🧩

#Practice Questions

Here are a few practice questions to get you warmed up:

Practice Question

#Multiple Choice Questions

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

2. A ball is thrown vertically upward. Which of the following is true about the ball's velocity and acceleration at its highest point? (A) Both velocity and acceleration are zero. (B) Velocity is zero, but acceleration is nonzero. (C) Velocity is nonzero, but acceleration is zero. (D) Both velocity and acceleration are nonzero.

3. A projectile is launched at an angle. If air resistance is negligible, which of the following remains constant during the projectile’s motion? (A) Vertical velocity (B) Horizontal velocity (C) Vertical acceleration (D) Total velocity

#Free Response Question

A small block of mass $m$ is released from rest at the top of a frictionless ramp of height $h$ and angle $\theta$. At the bottom of the ramp, it slides onto a horizontal surface with friction, eventually coming to rest after traveling a distance $d$.

(a) Derive an expression for the speed of the block at the bottom of the ramp in terms of $g$, $h$, and $\theta$. (b) Derive an expression for the acceleration of the block on the horizontal surface in terms of $g$, $m$, and the coefficient of kinetic friction $\mu_k$. (c) Derive an expression for the distance $d$ the block travels on the horizontal surface in terms of $g$, $h$, $\theta$, and $\mu_k$.

#Scoring Breakdown

(a) Speed at the bottom of the ramp (3 points)

• 1 point for using conservation of energy or kinematics

• 1 point for correct potential energy at the top: $mgh$

• 1 point for correct kinetic energy at the bottom: $\frac{1}{2}mv^2$

  $mgh = \frac{1}{2}mv^2$

  $v = \sqrt{2gh}$

(b) Acceleration on the horizontal surface (3 points)

• 1 point for recognizing that friction is the net force

• 1 point for correct friction force: $f_k = \mu_k mg$

• 1 point for using Newton's second law: $F = ma$

  $-\mu_k mg = ma$

  $a = -\mu_k g$

(c) Distance traveled on the horizontal surface (4 points)

• 1 point for using kinematics with the correct initial velocity

• 1 point for using the correct final velocity (0)

• 1 point for using the correct acceleration from part (b)

• 1 point for correct final answer

  $v_f^2 = v_i^2 + 2ad$

  $0 = 2gh + 2(-\mu_k g)d$

  $d = \frac{h}{\mu_k}$

You've got this! Go crush that exam! 🌟