Projectile Motion Vocabulary

Units of Measurement and System Conversions

Overview of Measurement Systems

• Metric System (International System of Units, SI): The standard system of measurement used by most countries worldwide.

• English System (Imperial or US Customary System): The system of measurement commonly used in the United States.

Measurement System Comparison

• Length:

  • SI (Metric): Millimeter ($\text{mm}$), Centimeter ($\text{cm}$), Meter ($\text{m}$), Kilometer ($\text{km}$).

  • English: Inch ($\text{in}$), Foot ($\text{ft}$), Yard ($\text{yd}$), Mile ($\text{mi}$).

• Mass / Weight:

  • SI (Metric): Milligram ($\text{mg}$), Gram ($\text{g}$), Kilogram ($\text{kg}$).

  • English: Ounce ($\text{oz}$), Pound ($\text{lb}$), Ton ($\text{T}$).

• Capacity / Volume:

  • SI (Metric): Milliliter ($\text{mL}$), Liter ($\text{L}$), Kiloliter ($\text{kL}$).

  • English: Fluid Ounce ($\text{fl oz}$), Cup ($\text{c}$), Pint ($\text{pt}$), Quart ($\text{qt}$), Gallon ($\text{gal}$).

• Temperature:

  • SI (Metric): Degree Celsius ($^\circ\text{C}$), Kelvin ($\text{K}$).

  • English: Degree Fahrenheit ($^\circ\text{F}$).

Unit Conversions

Converting Within the English System

• Dimensional Analysis Rule: Place the unit you need in the numerator and the given unit in the denominator.

• Conversion Constant: $1\,\text{kg} = 2.21\,\text{lbs}$.

• Example Setup: Converting $2.5\,\text{kg}$ yields $2.5 \times 2.21 = 5.525\,\text{lbs}$.

Converting Within the Metric System

• Metric prefixes represent multiples or fractions of a base unit ($\text{m}$, $\text{g}$, $\text{L}$).

• Metric Prefix Chart (KHD B DCM):

  • Kilo- ($\text{k}$): Multiplier value $10^{-3}$ relative scale ($1000$ base units).

  • Hecto- ($\text{h}$): Multiplier value $10^{-2}$ relative scale ($100$ base units).

  • Deka- ($\text{da}$): Multiplier value $10^{-1}$ relative scale ($10$ base units).

  • Base Unit: ($\text{m}$, $\text{g}$, $\text{L}$).

  • Deci- ($\text{d}$): Multiplier value $10^1$ relative scale ($0.1$ base unit).

  • Centi- ($\text{c}$): Multiplier value $10^2$ relative scale ($0.01$ base unit).

  • Milli- ($\text{m}$): Multiplier value $10^3$ relative scale ($0.001$ base unit).

Metric Conversion Practice Problems

• Problem 1: A bottle contains $4500\,\text{mL}$ of juice. Express the capacity in liters ($\text{L}$).

  • Solution: $4500\,\text{mL} \div 1000 = 4.5\,\text{L}$.

• Problem 2: A wire measures $950\,\text{cm}$ in length. Express the length in meters ($\text{m}$).

  • Solution: $950\,\text{cm} \div 100 = 9.5\,\text{m}$.

• Problem 3: A laboratory sample has a mass of $8.75\,\text{kg}$. Express the mass in grams ($\text{g}$).

  • Solution: $8.75\,\text{kg} \times 1000 = 8750\,\text{g}$.

Forces and Newton's Laws of Motion

Fundamental Concepts of Force

• Definition of Motion: The observable change in position relative to a reference point.

• Definition of Force: A push or a pull in a particular direction resulting from an object's interaction with another object.

• Pulling: The force exerted to move something closer to oneself (e.g., pulling a block causes it to move toward the user).

• Functions of Force: Force can change an object's speed, direction, or state of rest.

Newton's First Law of Motion: Law of Inertia

• Statement: An object at rest stays at rest, and an object in motion stays in motion unless acted upon by an unbalanced external force.

• Inertia: The tendency of an object to resist changes in its state of motion.

• Mass Relationship: Mass is a direct measure of inertia; the greater the mass, the greater the inertia.

Newton's Second Law of Motion: Law of Acceleration

• Statement: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

• Mathematical Formulas:

  • Force: $F = m \times a$

  • Mass: $m = \frac{F}{a}$

  • Acceleration: $a = \frac{F}{m}$

• Units: Force is measured in Newtons ($\text{N}$), where $1\,\text{N} = 1\,\text{kg}\,\text{m/s}^2$

• Variable Dynamics:

  • Force vs. Acceleration: Pushing a cart harder (more force) causes it to speed up faster (more acceleration).

  • Mass vs. Acceleration: Filling a cart with groceries (more mass) requires more force to achieve the same acceleration.

Newton's Third Law of Motion: Law of Interaction

• Statement: For every action, there is an equal and opposite reaction.

• Action-Reaction Pairs: Forces always occur in simultaneous, equal, and opposite pairs.

Primary Categories of Forces

• Contact Forces: Result from physical contact between two touching objects.

  • Examples: Dribbling a basketball, kicking a soccer ball, hitting a shuttlecock.

• Non-Contact Forces: Act on an object without coming into direct physical contact.

  • Magnetic Force: The push or pull exerted by a magnet that attracts or repels magnetic objects.

  • Gravitational Force: The force by which an object attracts another object towards itself (e.g., Earth pulling objects downward).

  • Electrostatic Force: The force existing between electrically charged particles.

Free Body Diagrams (FBD)

• Definition: A graphic representation of all external forces acting upon an object or system.

• Purpose: Helps visualize physical problems and streamlines problem-solving.

• Standardized Force Symbols:

  • Gravitational Force: $F_{\text{grav}}$

  • Normal Force: $F_{\text{norm}}$

  • Applied Force: $F_{\text{app}}$

  • Frictional Force: $F_{\text{fric}}$

  • Tension Force: $F_{\text{tens}}$

  • Air Resistance: $F_{\text{air}}$

• Construction Rules:

  • All force arrows must originate at the object and point outward in the direction the force acts.

• Tug of War Practice Scenario:

  • Scenario: Six players divided equally into two groups playing tug of war, with both groups exerting a force of $375\,\text{N}$ on each side.

  • Net Force Calculation: $F_{\text{net}} = 375\,\text{N} - 375\,\text{N} = 0\,\text{N}$.

  • Force State: Balanced (net force is zero).

One-Dimensional Kinematics

Kinematic Concepts and Definitions

• Kinematics: The classification and comparison of motions.

• Core Kinematic Restrictions:

  1. Motion occurs along a straight line only.

  2. Focuses exclusively on the motion itself without considering forces.

  3. The moving object is modeled as a point particle.

Distance versus Displacement

• Distance ($d$):

  • Scalar quantity (magnitude only).

  • SI Unit: meters ($\text{m}$).

  • Represents the complete length of the path traveled by an object from start to finish.

  • Example: Jogging $400\,\text{m}$ around a track yields a total distance of $400\,\text{m}$.

• Displacement ($d$):

  • Vector quantity (magnitude and direction).

  • SI Unit: meters ($\text{m}$).

  • Represents the shortest straight-line distance measured from the initial to the final position.

  • Resultant Displacement ($d_R$): The vector sum of all individual displacement vectors.

• Sign Conventions:

  • North and East directions: Positive ($+$).

  • South and West directions: Negative ($-$).

Walk Trajectory Practice Problems

• Scenario 1: Alex walks $10\,\text{m}$ East, then $10\,\text{m}$ West.

  • Total Distance: $d_T = 10\,\text{m} + 10\,\text{m} = 20\,\text{m}$.

  • Resultant Displacement: $d_R = (+10\,\text{m}) + (-10\,\text{m}) = 0\,\text{m}$.

• Scenario 2: Alex walks East $10\,\text{m}$, South $10\,\text{m}$, East $10\,\text{m}$, South $10\,\text{m}$, West $15\,\text{m}$, North $10\,\text{m}$, West $5\,\text{m}$.

  • Horizontal Components: $d_x = (+10\,\text{m}) + (+10\,\text{m}) + (-15\,\text{m}) + (-5\,\text{m}) = 0\,\text{m}$.

  • Vertical Components: $d_y = (-10\,\text{m}) + (-10\,\text{m}) + (+10\,\text{m}) = -10\,\text{m}$ ($10\,\text{m}$ South).

  • Total Distance: $d_T = 10\,\text{m} + 10\,\text{m} + 10\,\text{m} + 10\,\text{m} + 15\,\text{m} + 10\,\text{m} + 5\,\text{m} = 70\,\text{m}$.

  • Resultant Displacement: $d_R = 10\,\text{m}$ South.

Speed versus Velocity

• Speed:

  • Measure of how fast an object moves (distance traveled per unit time).

  • Scalar quantity (magnitude only).

  • Core Formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

  • Derived Equations:

  • $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

  • $\text{Distance} = \text{Speed} \times \text{Time}$

  • Units: Units of distance divided by units of time (e.g., $\text{m/s}$, $\text{miles/hour}$ or $\text{mph}$).

  • Example: Traveling $100\,\text{m}$ in $10\,\text{s}$ gives a speed of $10\,\text{m/s}$. A car moving at $60\,\text{km/h}$.

• Velocity:

  • Speed with direction included; tells how fast and where an object is moving.

  • Vector quantity (magnitude and direction).

  • Core Formula: $\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$

  • Example: $5\,\text{m/s}$ East; A car moving at $60\,\text{km/h}$ North.

• Practical Significance: Understanding speed and velocity is crucial in transportation, sports, science, and engineering.

Kinematic Equations for Straight-Line Motion

• General Kinematic Formulas:

  • $v_f = v_i + a t$

  • $v_f^2 = v_i^2 + 2 a d$

  • $d = v_i t + \frac{1}{2} a t^2$

  • $d = \left(\frac{v_f + v_i}{2}\right) t$

• Horizontal Motion vs. Vertical Free Fall:

  • Horizontal Motion:

  • Horizontal distance ($d_x$): Measured in meters ($\text{m}$); $d_x = v_i t + \frac{1}{2} a t^2$

  • Horizontal acceleration ($a$): Measured in $\text{m/s}^2$; $a = \frac{v_f - v_i}{t}$

  • Initial velocity ($v_i$) and Final velocity ($v_f$): Measured in $\text{m/s}$; $v_f = v_i + a t$

  • Time ($t$): Measured in seconds ($\text{s}$); $t = \frac{d}{v_i}$

  • Vertical Motion (Free Fall under Gravity):

  • Vertical distance ($d_y$ or $h$): Measured in meters ($\text{m}$); $d_y = \frac{1}{2} g t^2$

  • Acceleration due to gravity ($g$): $g = -9.8\,\text{m/s}^2$

  • Horizontal acceleration component ($a_x$): $a_x = 0\,\text{m/s}^2$

  • Final vertical velocity ($v_f$): Measured in $\text{m/s}$; $v_f = v_i + g t$

Horizontal Motion Sample Problem

• Problem: A car is moving horizontally with an initial velocity of $10\,\text{m/s}$ and accelerates at $2\,\text{m/s}^2$ for $5\,\text{s}$. What horizontal distance does it travel?

• Given: $v_i = 10\,\text{m/s}$, $a = 2\,\text{m/s}^2$, $t = 5\,\text{s}$

• Formula: $d = v_i t + \frac{1}{2} a t^2$

• Calculation:

  • $d = (10\,\text{m/s})(5\,\text{s}) + \frac{1}{2}(2\,\text{m/s}^2)(5\,\text{s})^2$

  • $d = 50\,\text{m} + \frac{1}{2}(2\,\text{m/s}^2)(25\,\text{s}^2)$

  • $d = 50\,\text{m} + 25\,\text{m} = 75\,\text{m}$

• Answer: $d = 75\,\text{m}$

Two-Dimensional Kinematics: Projectile Motion

Fundamentals of Projectile Motion

• Definition: The motion of a body launched horizontally or at an angle other than $90^\circ$ with the horizontal; motion of an object thrown or projected into the air that moves under constant acceleration.

• Core Terminology:

  • Projectile: The body or object being thrown upon which the only force acting is gravity.

  • Trajectory: The curved path followed by a projectile; it forms a parabolic path.

  • Range: The horizontal distance ($x$ or $d_x$) traveled by the projectile between its launching point and landing point.

  • Projectile Motion: The motion of a body thrown in a curved path under constant acceleration.

• Launch Classifications:

  1. Launched horizontally.

  2. Launched at an angle.

Horizontally Launched Projectiles

• Characteristics:

  • Has NO upward trajectory and NO initial vertical velocity ($v_{0y} = 0\,\text{m/s}$ or $v_{iy} = 0\,\text{m/s}$).

  • Horizontal velocity remains constant ($v_{ix} = v_{fx} = \text{constant}$).

  • Vertical motion is free fall ($v_{iy} = 0\,\text{m/s}$) with constant acceleration due to gravity ($a_y = g = -9.8\,\text{m/s}^2$).

  • A combination of uniform horizontal velocity and free-fall vertical motion.

  • Gravity accelerates objects downward but is unable to affect horizontal motion, resulting in a curved path.

Component Breakdown of Projectile Motion

• Horizontal Component (Subscript $x$):

  • Motion Type: Uniform motion.

  • Acceleration: $a_x = 0\,\text{m/s}^2$ (constant velocity).

  • Velocity: $v_{fx} = v_{ix}$ (or $v_{ix} = v_i \cos(\theta)$).

  • Displacement (Range): $x = v_{ix} t$ or $d_x = v_x t$.

• Vertical Component (Subscript $y$):

  • Motion Type: Free fall (body dropped).

  • Acceleration: $a_y = g = -9.8\,\text{m/s}^2$ (constant vertical acceleration).

  • Initial Velocity: $v_{iy} = 0\,\text{m/s}$ for horizontal launch (or $v_{iy} = v_i \sin(\theta)$ for angled launch).

  • Final Velocity: $v_{fy} = v_{iy} + g t$

  • Displacement (Height $y$ or $d_y$): $y = v_{iy}t + \frac{1}{2} g t^2$, or $v_{fy}^2 = v_{iy}^2 + 2 g y$

• Time Integration: Horizontal and vertical time are equal ($t_x = t_y = t$).

• Instantaneous Velocity Magnitude:

  • Initial Magnitude: $v_i = \sqrt{v_{ix}^2 + v_{iy}^2}$

  • Final Magnitude: $v_f = \sqrt{v_{fx}^2 + v_{fy}^2}$

Horizontally Launched Projectile Practice Problems

Sample Problem 1: Rolling Marble off Table Edge

• Problem Description: A marble rolled on top of a table at constant velocity and off the table's edge unto the floor. It took the marble $0.5\,\text{s}$ to hit the floor from the moment it left the table's edge. The marble landed $1\,\text{m}$ from the point directly below the edge of the table.

• Given Parameters:

  • Horizontal displacement $x = 1\,\text{m}$

  • Time $t = 0.5\,\text{s}$

  • Initial vertical velocity $v_{iy} = 0\,\text{m/s}$

  • Vertical acceleration due to gravity $g = -9.8\,\text{m/s}^2$

• Part (a): Find the height of the table ($y$)

  • Formula: $y = v_{iy}t + \frac{1}{2} g t^2$

  • Solution:

  • $y = (0\,\text{m/s})(0.5\,\text{s}) + \frac{1}{2}(-9.8\,\text{m/s}^2)(0.5\,\text{s})^2$

  • $y = 0 + \frac{1}{2}(-9.8\,\text{m/s}^2)(0.25\,\text{s}^2)$

  • $y = -1.225\,\text{m} \approx -1.23\,\text{m}$

  • Answer: $y = -1.23\,\text{m}$ (the negative sign indicates downward movement, corresponding to a table height of $1.23\,\text{m}$).

• Part (b): Find the rolling velocity on the table ($v_{ix}$)

  • Formula: $x = v_{ix} t \implies v_{ix} = \frac{x}{t}$

  • Solution:

  • $1\,\text{m} = v_{ix} (0.5\,\text{s})$

  • $v_{ix} = \frac{1\,\text{m}}{0.5\,\text{s}} = 2.0\,\text{m/s}$

  • Answer: $v_{ix} = 2.0\,\text{m/s}$

Sample Problem 2: Rescue Plane Emergency Supply Drop

• Problem Description: A person is trapped in the snow, and a rescue plane flying horizontally at an altitude of $5000\,\text{m}$ at a speed of $500\,\text{m/s}$ drops emergency supplies (assuming no air resistance).

• Given Parameters:

  • Vertical displacement $y = -5000\,\text{m}$

  • Initial horizontal velocity $v_{ix} = 500\,\text{m/s}$

  • Initial vertical velocity $v_{iy} = 0\,\text{m/s}$

  • Acceleration due to gravity $g = -9.8\,\text{m/s}^2$

• Part (a): Calculate the time ($t$) required for supplies to hit the ground

  • Formula Derivation: $y = \frac{1}{2} g t^2 \implies t = \sqrt{\frac{2y}{g}}$

  • Solution:

  • $t = \sqrt{\frac{2(-5000\,\text{m})}{-9.8\,\text{m/s}^2}}$

  • $t = \sqrt{\frac{-10000}{-9.8}} = \sqrt{1020.408} \approx 31.94\,\text{s}$

  • Answer: $t = 31.94\,\text{s}$

• Part (b): Calculate how far in front of the person ($x$) the pilot should drop the supplies

  • Formula: $x = v_{ix} t$

  • Solution:

  • $x = (500\,\text{m/s})(31.94\,\text{s})$

  • $x = 15971.91\,\text{m}$

  • Answer: $x = 15971.91\,\text{m}$