Kinematics of Translation (General Physics 1)

• Kinematics of Translation (General Physics 1, Lesson 1.3) – Key learning targets

  • Convert a verbal description of a physical situation involving uniform acceleration in one dimension into a mathematical description
  • Describe motion using the concept of relative velocities in one and two dimensions
  • Solve problems in one-dimensional motion including free-fall
  • Extend the definition of position, velocity, and acceleration to 2D and 3D using vector representation
  • Differentiate uniform from non-uniform circular motion
  • Infer quantities associated with circular motion
  • Solve problems in two or three dimensions by resolving motion into components
  • Construct motion graphs and determine displacement, velocity, and acceleration from these graphs
  • Plan and execute an experiment involving projectile motion

• Boxed concepts in this topic:

  • Motion
  • Uniform displacement
  • Velocity graphs
  • Equations of motion
  • Calculus
  • Acceleration
  • Uniformly accelerated motion
  • Free fall
  • Projectile motion

• Kinematic vs Dynamic concepts

  • Kinematics describes motion in terms of displacement, velocity, and acceleration
  • Dynamics studies force in relation to motion
  • Translation: motion in a straight line

• Displacement and Distance

  • Distance: total length of path from initial to final position
  • Displacement: straight-line distance from initial to final position, with direction toward the final position
  • Mathematical expression: riangle x = x - x_0

• Example 1 – Route distance and displacement

  • Route: 50 m north, 40 m east, 60 m north
  • Return path: same route in opposite directions
  • (a) Total distance traveled: 50 + 40 + 60 + 60 + 40 + 50 = 300 ext{ m}
  • (b) Total displacement: Net displacement after completing the round trip is zero (back to the start)

• Speed vs Velocity; Average vs Instantaneous

  • Speed: distance traveled per unit time; scalar
  • Velocity: displacement per unit time; vector
  • Equations:
  • Speed: v = rac{dx}{dt} ext{? (dist. travelled) } [distances are interpreted as total distance over time]
  • Velocity: oldsymbol{v} = rac{doldsymbol{x}}{dt}
  • Instantaneous vs Average:
  • Average speed: v_{ ext{avg}} = rac{ ext{total distance}}{ ext{total time}}
  • Instantaneous speed: the speed at a particular moment
  • Average velocity: oldsymbol{v}{ ext{avg}} = rac{oldsymbol{x} - oldsymbol{x}0}{t - t_0} = rac{ riangle oldsymbol{x}}{ riangle t}
  • Instantaneous velocity: oldsymbol{v}{ ext{inst}} = abla{ ext{t}} oldsymbol{x} = rac{doldsymbol{x}}{dt}

• Relative velocity

  • The velocity of a moving body may differ when viewed from different frames of reference
  • Example: a sidewalk problem (moving walkway) where velocity relative to ground is the sum of sidewalk speed and walker speed relative to sidewalk

• Example 2 – Pursuit with different speeds

  • Given: Vania travels 360 m at 3.0 m/s; 10 s later Angelito travels at 4.0 m/s
  • Tasks (a) time for Angelito to overtake Vania; (b) distance from school when overtaken
  • Concept: relative motion along the same route

• Example 3 – Moving sidewalk (airport terminal)

  • Sidewalk length = 50.0 m; sidewalk speed = 1.25 m/s
  • (a) If the lady is standing still on the sidewalk, time to reach end: t = rac{L}{v_{ ext{sidewalk}}} = rac{50.0}{1.25} = 40.0 ext{ s}
  • (b) If she walks at 1.50 m/s relative to the sidewalk in the same direction, ground speed = v = v{ ext{sidewalk}} + v{ ext{walker}} = 1.25 + 1.50 = 2.75 ext{ m/s}; time = t = rac{50.0}{2.75} \approx 18.18 ext{ s}

• Acceleration; instantaneous vs average

  • Acceleration definition: a = rac{ riangle v}{ riangle t} = rac{v - v_0}{t}
  • Instantaneous acceleration: a_{ ext{inst}} = rac{dv}{dt}
  • Average velocity in uniformly accelerated motion: oldsymbol{v}{ ext{avg}} = rac{oldsymbol{v} + oldsymbol{v0}}{2}

• Example 4 – A time-dependent position function

  • Given: x(t) = 3.5 ext{ m s}^3 rac{t^3}{?} - 2 ext{ m s} rac{t}{} + 5.0 ext{ m}
  • Velocity: v(t) = rac{dx}{dt} = 10.5 t^2 - 2
  • Acceleration: a(t) = rac{dv}{dt} = 21 t
  • At t = 2.0 s: v(2) = 10.5(4) - 2 = 40 ext{ (units as given)}; \ a(2) = 21(2) = 42

• Uniformly Accelerated Motion – Other Kinematic Equations (SUVAT)

  • Standard relations (assuming constant acceleration a):
  • v = v_0 + a t
  • x = x0 + v0 t + rac{1}{2} a t^2
  • v^2 = v0^2 + 2 a (x - x0)
  • x = rac{v + v_0}{2} t

• Example 5 – Two successive time intervals with uniform acceleration

  • Given: a uniformly accelerated car covers 250 m in t1 = 5.0 s, then another 250 m in t2 = 3.0 s
  • Solve for acceleration a (and initial velocity v0):
  • First interval: 250 = v0 (5.0) + rac{1}{2} a (5.0)^2 = 5 v0 + 12.5 a
  • Velocity at t1: v1 = v0 + a (5.0)
  • Second interval: 250 = v1 (3.0) + rac{1}{2} a (3.0)^2 = 3(v0 + 5a) + 4.5 a = 3 v_0 + 19.5 a
  • Solve: subtracting gives v_0 = 3.5 a; substitute into first equation: 250 = 5(3.5 a) + 12.5 a = 30 a
    ightarrow a \n = rac{250}{30}
    \approx 8.33 ext{ m s}^{-2}
  • Then v_0

ightarrow v_0 = 3.5 a

• Practical note – Tailgating and safe following distance
  • Tailgating: driving too closely behind another vehicle; safety concerns
  • Recommended safe following distance: minimum of three seconds between the back of the lead vehicle and the front of your vehicle
  • How to apply: pick a fixed marker; ensure you pass it at least three seconds after the lead vehicle passes
  • Relationship with speed: faster speeds require greater following distance
• Free Fall: fundamental concepts
  • In vacuum (no air resistance), all bodies fall with the same acceleration near Earth's surface, independent of mass
  • Acceleration due to gravity: g \,=\, -9.81\ \text{m s}^{-2} (downward direction toward Earth’s center)
  • Direction: downward is negative in the common convention; upward velocity is positive; downward velocity is negative
  • Variation of g with location: g is approximately constant but does vary slightly by location
• Misconceptions about Free Fall
  • Heavier objects do not fall faster than lighter ones (in absence of air resistance)
  • Acceleration is independent of mass
  • Motion does not require an applied force to continue; once in motion, velocity changes due to gravity
  • Constant speed is not maintained in free fall unless there is an opposing force (air resistance)
• Free-Fall equations (vertical motion under constant gravity)
  • Velocity: vy = v{0y} + a_y t
  • Position: y = y0 + v{0y} t + frac{1}{2} a_y t^2
  • With downward positive convention (or modify signs accordingly); for standard convention with downward acceleration g > 0, you may see a_y = -g and the equations adapt accordingly
• Example 6 – Vertical throw from height
  • Problem: A stone is thrown upward with initial velocity v_{0y} = 4.9\,\text{m/s} from a 52 m high building; it returns to ground below the building top
  • (a) Maximum height relative to ground
  • (b) Time of flight (total time in air)
  • (c) Velocity just before reaching the ground
  • Results (approx):
  • Maximum height above ground: y{\max} = 52 + \frac{v{0y}^2}{2g} = 52 + \frac{(4.9)^2}{2\cdot 9.81} \approx 53.22\ \text{m}
  • Time of flight: Solve 0 = 52 + v{0y} t - \tfrac{1}{2} g t^2; numerical solution gives t{\text{total}} \approx 3.80\ \text{s}
  • Velocity just before ground: vy = v{0y} - g t_{\text{total}} \approx 4.9 - 9.81(3.80) \approx -32.3\ \text{m/s} (downward)
• Motion in More Than One Dimension
  • Real-world motion is not limited to one dimension; decompose into components
  • Resolve motion into its straight-line components along each axis (e.g., x and y)
• Derivatives of Position (math framework)
  • Position: \mathbf{r}(t)
  • Velocity: \mathbf{v} = \frac{d\mathbf{r}}{dt}
  • Acceleration: \mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2 \mathbf{r}}{dt^2}
  • Jerk: \mathbf{j} = \frac{d\mathbf{a}}{dt} = \frac{d^3 \mathbf{r}}{dt^3}
  • Higher derivatives: snap, crackle, pop (as listed: jerk, snap, crackle, pop)
• Example 7 – 2D motion under a perpendicular force
  • Initial motion: along x-axis at v_x = 2.6\ \,\text{m/s}
  • A constant force acts along y-axis, causing acceleration a_y = 3.8\ \,\text{m/s}^2 for 5.0 s
  • (a) Position after 5.0 s:
  • x = v_x t = 2.6 \times 5.0 = 13.0\ \text{m}
  • y = \tfrac{1}{2} a_y t^2 = \tfrac{1}{2} (3.8) (5.0)^2 = 0.5 \times 3.8 \times 25 = 47.5\ \text{m}
  • (b) Velocity after 5.0 s:
  • v_x = 2.6\ \text{m/s} (constant)
  • vy = ay t = 3.8 \times 5.0 = 19.0\ \text{m/s}
  • Resultant speed: |\mathbf{v}| = \sqrt{vx^2 + vy^2} \approx \sqrt{2.6^2 + 19^2} \approx 19.2\ \text{m/s}
  • Direction: angle relative to +x, \theta = \tan^{-1}\left( \frac{vy}{vx} \right) \approx \tan^{-1}(\tfrac{19}{2.6}) \approx 82.2^{\circ}
• Projectile Motion – Core ideas
  • A projectile is an object thrown horizontally or at an angle; examples include soccer balls, basketballs, bullets
  • Projectile motion is a combination of a constant horizontal motion and a vertical motion under gravity (free fall in the vertical component)
  • The horizontal and vertical motions are independent
  • Vertical motion shows time and speed symmetries along the vertical component
• Types of Projectile Motion
  • Horizontal: projection angle θ = 0°
  • Vertical: projection angle θ = 90°
  • General case: 0° < θ < 90°
• Factors affecting projectile motion
  • Gravity (g), projection angle (θ), air resistance, initial speed, initial height
  • Example ranges and heights depend on these factors
• Vertical component of projectile motion
  • Vertical acceleration due to gravity; velocity changes with time; at the top, vertical velocity is zero but acceleration remains g downward
  • Projectile motion is a combination of vertical motion under g and horizontal motion with no horizontal force (a_x = 0)
• Complementary angles in projectile motion
  • For the same initial speed, the same range can be achieved with two projection angles that are complementary: θ and 90° − θ
• Components of projectile motion (summary)
  • Horizontal: x = R = vx t; vx = vi cos θ; ax = 0
  • Vertical: y = v{iy} t + (1/2) g t^2; vy = v{iy} + g t; ay = g (downward)
  • Maximum height occurs when vy = 0; at that instant t = v{iy} / g
• Finding velocity in horizontal and vertical motion
  • Horizontal velocity: constant, vx = vi \cos\theta
  • Vertical velocity: vy(t) = vi \sin\theta - g t (sign convention depends on upward positive)
• Circular Motion
  • Centripetal acceleration: a_c = \frac{v^2}{r} directed toward the center
  • Tangential speed: v = \frac{2\pi r}{T} where T is the period
  • Direction of motion is tangential to the circle
  • Relationship with radius: larger radius with the same period implies larger speed
• Example 10 – Merry-go-round
  • Given: radius r = 3.0 m; rotation rate: 2 revolutions in 5.0 s
  • Period: T = \frac{5.0\ \text{s}}{2} = 2.5\ \text{s}
  • Tangential speed: v = \frac{2\pi r}{T} = \frac{2\pi \times 3.0}{2.5} \approx 7.54\ \text{m/s}
  • Centripetal acceleration: a_c = \frac{v^2}{r} = \frac{(7.54)^2}{3.0} \approx 18.9\ \text{m/s}^2
• Graphical analysis of motion
  • Slope interpretation (Position vs Time): slope = velocity
  • Slope sign indicates direction of motion
  • Distance traveled corresponds to area under Velocity vs Time graphs
• Position vs Time graphs (typical shapes)
  • Uniform motion: slope constant (positive) with a = 0
  • Uniformly accelerated motion: slope increasing linearly with time; a = constant (>0)
  • Uniformly decelerated motion: slope decreasing (negative acceleration)
  • Non-uniformly accelerated motion: curvature in the graph
• Velocity vs Time graphs
  • Flat line: constant speed (a = 0)
  • Positive slope: increasing velocity (positive acceleration)
  • Negative slope: decreasing velocity (negative acceleration or deceleration)
  • Area under the curve = distance traveled (for one-dimensional motion)
• Acceleration vs Time graphs
  • Constant acceleration vs non-uniform acceleration
• Real-world context notes
  • Jeepneys and public transport – real-world relevance of kinematics in transportation and design choices
  • Kinematics as a foundation for physics in technology and gaming (e.g., projectile trajectories in Angry Birds)
• Performance Task (P Task)
  • A culminating activity to apply the concepts learned in kinematics to a practical problem
• Recap of key formulas to memorize
  • Displacement: \triangle x = x - x_0
  • Velocity (average): \boldsymbol{v}{avg} = \frac{\boldsymbol{x} - \boldsymbol{x}0}{t - t_0}
  • Velocity (instantaneous): \boldsymbol{v}_{inst} = \frac{d\boldsymbol{x}}{dt}
  • Speed (average): v_{avg} = \frac{\text{distance}}{\text{time}}
  • Acceleration: \mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2}
  • SUVAT (constant acceleration):
  • v = v_0 + a t
  • x = x0 + v0 t + \tfrac{1}{2} a t^2
  • v^2 = v0^2 + 2 a (x - x0)$$
• Connections to foundational ideas
  • Kinematics connects to vectors, calculus, and geometry via rate-of-change concepts
  • Relative motion and frame-of-reference analysis ties into basic physics principles of observer-dependence in measurements
  • Graphical analysis provides intuitive visualization of how displacement, velocity, and acceleration relate over time
• Practical implications and ethics
  • Understanding safe following distances reduces accidents (tailgating discussion)
  • Accurate modeling of projectile and circular motion informs design of machinery, vehicles, sports equipment, amusement rides, and simulations
  • Recognizing limits of models (e.g., neglecting air resistance in free-fall or projectile analyses) is essential for real-world engineering and safety considerations
• Notable historical/contextual notes from the slides
  • Ivan Getting and GPS development – historical context for advancements in science and technology
  • Geopolitical and technological impact of science (NASA, GPS, aerospace programs) as part of the broader story of physics applications
• Summary for exam prep
  • Be able to translate verbal motion descriptions into equations for 1D motion, including uniform acceleration and free fall
  • Resolve 2D/3D motion into components along orthogonal axes and analyze each component separately
  • Derive and apply kinematic equations for constant acceleration (SUVAT) and use them to solve problems
  • Analyze projectile motion via horizontal and vertical components, including time of flight, range, maximum height, and velocity vectors
  • Interpret and construct position-time, velocity-time, and acceleration-time graphs, including understanding areas and slopes
  • Understand relative velocity and frames of reference, including practical examples like moving sidewalks and pursuit problems
  • Recognize common misconceptions and correct them with the physics model
• Note: Some slide content includes ancillary material (e.g., social/educational memes, unrelated company notes, etc.). The core physics concepts of kinematics are captured above and aligned with the students’ learning targets and examples included in the transcript.