Projectile Motion & Forces

Projectile Motion

Basic Concepts

• Kinematic Equations: These equations are used for each component of motion (x and y). Time is the only common variable between them.
• 2D & 3D Kinematic Equations:
  • x-component: $x<em>f = \frac{1}{2} a</em>x t^2 + v<em>{i,x} t + x</em>i$
  • y-component: $y<em>f = \frac{1}{2} a</em>y t^2 + v<em>{i,y} t + y</em>i$
  • x-velocity: $v<em>{f,x} = a</em>x t + v_{i,x}$
  • y-velocity: $v<em>{f,y} = a</em>y t + v_{i,y}$
  • x-velocity squared: $v<em>{f,x}^2 = v</em>{i,x}^2 + 2 a_x \Delta x$
  • y-velocity squared: $v<em>{f,y}^2 = v</em>{i,y}^2 + 2 a_y \Delta y$
  • Vector form: $\vec{r}<em>f = \frac{1}{2} \vec{a} t^2 + \vec{v}</em>i t + \vec{r}_i$
  • Velocity vector: $\vec{v}<em>f = \vec{a} t + \vec{v}</em>i$

Clicker Question 9

• Question: If a ball is released from rest and another is shot horizontally from the same height, which hits the ground first?
• Answer: C) Both hit at the same time.
• Explanation: Vertical motion is independent of horizontal motion. Video explanation at: https://www.youtube.com/watch?v=zMF4CD7i3hg (at 1:00)

Gravitational Field

• Direction: Downward.
• Magnitude: $g = 9.8 \frac{m}{s^2}$ near Earth’s surface.
• Decreases with distance from Earth’s center.
• $g$ is four times smaller if twice as far from the center.

Projectile Motion Details

• Basic Idea: Free fall with horizontal motion.
• Assumptions:
  • Negligible air resistance.
  • Not too fast or light.
  • Uniform gravitational field, $\vec{g}$.
  • Near Earth's surface.
  • Small trajectories compared to Earth.
• $\vec{g} \approx 9.8 \frac{m}{s^2}$

Projectile Motion Tools

• Acceleration components:
  • $\vec{a} = \vec{g}$
  • $a_x = 0$
  • $a_y = -g$
  • $g = 9.8 \frac{m}{s^2}$
• 2D kinematic equations with constant acceleration during free fall.

Trajectory

• The trajectory of projectile motion is a downward parabola.

Problem Solving Steps

1. Diagram the problem:
   • Include the trajectory.
   • Identify time intervals of constant acceleration.
   • Label moments where you know information.
   • Label moments where you want to know information.
2. List Knowns and Unknowns: At each significant moment.
3. Decide which kinematic equations will convert knowns into desired unknowns.

Problem Solving Tips

• Time relates horizontal and vertical motion (it's the only common variable).
• Unstated Information:
  • Max height implies $v_y = 0$
  • Known acceleration in projectile motion.
• Be careful with timing:
  • Differentiate between just before vs just after landing.
  • Velocity after hitting the ground is usually irrelevant.

Example Problem: Fire Fighting

• Scenario: A firefighter needs to spray water into a 1 m tall window 12 m above the ground. The nozzle is 1 m above the ground and 20 m from the building. If the water exits the nozzle at 35.0° above the horizontal at a speed of 30 m/s, will it enter the window?
• Solution: No, the water hits at 11.8 m above the ground, which is 0.2 m below the window.

Class Problem: Is it a Homerun?

• Scenario: A baseball is hit at a 40° angle from a height of 1 m above the ground. There is a 4 m tall wall 100 m away. Ignore air resistance.
• Question: How fast was the baseball moving right after it was hit to barely clear the wall?
• Answer: 32.1 m/s (approximately 72 mph).

Horizontal Range

• Requirements:
  • Same start and end height (level ground).
  • Free fall.
• Formula: $R = \frac{v<em>i^2 \sin 2\theta</em>i}{g}$
  • $R$ = Range
  • $v_i$ = Initial velocity
  • $\theta_i$ = Launch angle from horizontal
  • $g$ = Gravitational field magnitude

Maximum Height

• Basic Idea: How high the projectile reaches above launch height.
• Requirements: Free fall and level ground.
• Formula: $y<em>{max} = \frac{v</em>i^2 \sin^2 \theta_i}{2g}$
  • $y_{max}$ = Max height
  • $v_i$ = Initial velocity
  • $\theta_i$ = Launch angle from horizontal
  • $g$ = Gravitational field magnitude

Clicker Question 11

• Question: On Earth, the longest recorded golf drive was 471 m. About how far would it have gone on the moon, where gravity is 6 times weaker?
• Answer: C) 471(6) m

Example Problem: The Long Jump

• Scenario: A daredevil barely jumps over 24 trucks (each 3 m wide) off a 30° ramp.
• Question: How fast was he going when he left the ramp?
• Answer: 28.5 m/s or 63.8 mph.

Class Problem: Cannonball!

• Scenario: A cannon is fired at an angle of 60° across level ground. Neglecting air resistance, the cannonball leaves the barrel at 90 m/s.
• Questions:
  • How high does the cannonball reach?
  • How far does the cannonball go before hitting the ground?
• Answers:
  • Height: 310 m
  • Distance: 716 m

Circular Motion

Basic Concepts

• Easier to use directions relative to the circle's center.
• Definitions:
  • Centripetal ($c$): Toward the center.
  • Tangential ($t$): Counterclockwise along the circle.
• Use $c$ & $t$ for components like $x$ and $y$.

Circular Motion: Constant Speed

• Change in velocity is toward the center over short times.
• Acceleration is toward the center = “Centripetal” component, $a_c$.
• Formula: $a_c = \frac{v^2}{r}$
  • $a_c$ = Centripetal acceleration
  • $v$ = Speed
  • $r$ = Radius of the circle

Circular Motion: Changing Speed

• Turning requires centripetal acceleration, $a_c$ (how quickly velocity is turning).
• Changing speed requires tangential acceleration, $a_t$ (how quickly speed is changing).
• Direction:
  • Speeding up: $\vec{a}$ toward $\vec{v}$ (< 90°).
  • Slowing down: $\vec{a}$ opposite $\vec{v}$ (> 90°).
• Formulas:
  • $a_c = \frac{v^2}{r}$
  • $a<em>t = \frac{dv</em>t}{dt}$ (slope of tangent line on tangential component $v_t$ of velocity vs. time graph).

Circular Motion Variables

• Period, $T$: Time for a full rotation.
• Frequency, $f = \frac{1}{T}$: Number of cycles per second.
• Angular Speed, $\omega = 2\pi f$: Number of radians turned per second.
• These apply to constant speed cases.

Speed & Acceleration From Angular Variables

• Speed: $v = \frac{2\pi r}{T} = 2\pi f r = r \omega$
• Centripetal Acceleration: $a_c = \frac{v^2}{r} = \frac{4\pi^2 r}{T^2} = 4\pi^2 f^2 r = \omega^2 r$
• Requirements: Circular motion and constant speed.
• $r$ = radius of circle, $T$ = period, $f$ = frequency, $\omega$ = angular velocity

Forces

Introduction to Forces

• Rough Definition: Push and pull.
• Technical Definition: Interactions between two physical objects that pass the