Motion Notes: Displacement, Velocity, and Acceleration (Translational Motion and Free Fall)

Translational Motion: Core Concepts and Roadmap

• Motion in general can be decomposed into translational motion (center of mass movement) and rotational motion (about the center of mass).
• Start with point particles to develop intuition: translational motion is described without worrying about shape; rotational motion will be treated later for rigid bodies.
• The big three quantities introduced: displacement, velocity, and acceleration.
• Free fall is the central example of constant-acceleration motion near Earth’s surface when air resistance is neglected.
• Overview of the relationships among quantities will use graphs (position vs time, velocity vs time, acceleration vs time) and equations; both perspective are essential.

Displacement

• Displacement is the change in position between two instants, defined as \Delta x = xf - xi and is a vector quantity.
• Sign convention depends on the chosen coordinate axis: positive direction and origin must be specified.
• Displacement is not the same as distance traveled. Distance traveled is the total length of the path; displacement is the net change in position.
• Examples:
  • If you move from x=1 to x=3, displacement = $+2$.
  • If you move from x=3 to x=1, displacement = $-2$.
  • If you start at x=2 and return to x=2, displacement = 0 regardless of the distance traveled.
• Distance traveled can be larger than displacement; it is the sum of all leg lengths of the path.
• The sign and magnitude of displacement depend on the coordinate choice; it is the net change, not the total distance.

Coordinate Choice in 1D

• A 1D system uses a number line with an origin (zero) and a chosen positive direction.
• Position in 1D is often denoted by x (or y in vertical motion).
• R (position vector) is used in a more general multi-dimensional context; in 1D it essentially reduces to x.
• The origin and axis orientation are conventional and can be chosen to simplify problem solving (e.g., ground as origin for vertical motion).

Velocity: Average and Instantaneous

• Average velocity over a time interval is the slope of the displacement vs time over that interval:
  • \bar{v} = \frac{\Delta x}{\Delta t} = \frac{xf - xi}{tf - ti}
• Instantaneous velocity is the velocity at a specific time, i.e., the slope of the position vs time curve at that time (the tangent slope).
• If displacement is positive over a given interval, average velocity is positive; if negative displacement, velocity is negative.
• If displacement is zero (net no movement over the interval), average velocity is zero even if the object moved around in between.

Velocity-Time Graphs and Tangents

• A horizontal line in a velocity vs time graph indicates constant velocity (zero acceleration).
• A straight line in a velocity vs time graph with slope equal to the acceleration indicates constant acceleration.
• The slope of a velocity-time graph gives acceleration: a = \frac{\Delta v}{\Delta t}
• A non-straight velocity-time graph implies a changing velocity, i.e., nonzero acceleration that changes with time.
• Instantaneous velocity corresponds to the slope of the tangent to the velocity-time curve at that moment.

Acceleration: Definition and Units

• Acceleration is the rate of change of velocity with respect to time; for 1D motion:
  • a = \frac{\Delta v}{\Delta t}
• Units: [a] = \mathrm{m\,s^{-2}} (meters per second squared).
• Acceleration can be positive or negative depending on the sign convention; it is not simply equivalent to “speeding up” or “slowing down.”
• For constant acceleration, acceleration is a constant, so velocity changes linearly with time and position changes quadratically with time.

Constant Acceleration: Fundamental Kinematics

• With constant acceleration a, the basic kinematic equations (in a chosen coordinate system) are:
  • Velocity: v(t) = v_0 + a t
  • Position: x(t) = x0 + v0 t + \tfrac{1}{2} a t^2
  • Acceleration: a(t) = 0 (since a is constant)
• These equations imply: if a is constant, the position vs time is a parabola; velocity vs time is a straight line; acceleration vs time is a horizontal line.
• Many derived forms (no new physics, just algebraic rearrangements) are useful:
  • \Delta x = v_0 t + \tfrac{1}{2} a t^2 (displacement for a given time interval)
  • v^2 = v0^2 + 2 a (x - x0) (no time explicit; relates velocity to displacement)
  • x = x0 + \tfrac{v0 + v}{2} t (an average-velocity form, equivalent under certain manipulations)
• In problems you may be given any two of the variables (x, v, a, t) and asked to find the rest; you may also combine equations to eliminate a variable.

Free Fall Near Earth’s Surface

• Free fall is motion under gravity with air resistance neglected; gravity near Earth’s surface is treated as a constant acceleration.
• Standard gravitational acceleration is denoted by g; near the surface, g \approx 9.81\ \mathrm{m\,s^{-2}} (≈ 10 for quick hand calculations).
• Sign conventions:
  • If upward is chosen as positive, then gravity is a negative acceleration: a = -g.
  • If downward is chosen as positive, then gravity is positive: a = +g.
• For a freely falling object released from rest (initial velocity v_0 = 0) near the surface, with downward taken as positive:
  • Velocity grows as v(t) = g t after release.
  • Position decreases toward the ground following y(t) = y0 - \tfrac{1}{2} g t^2 (if y is measured upward from ground and downward is negative) or equivalently with downward positive, y(t) = y0 + \tfrac{1}{2} g t^2.
• Time to fall from height \Delta y (from rest, with downward positive) is t = \sqrt{ \frac{2 \Delta y}{g} }.
• Standard gravity affects all planets/moons; each body has its own surface gravity depending on mass and radius.

Parabolic Trajectories and Projectile Motion (Air Resistance Neglected)

• In projectile motion (2D) neglecting air resistance, horizontal velocity is constant and vertical velocity changes linearly due to gravity, producing a parabolic trajectory in the vertical plane.
• The center of mass follows the simplest path (a parabola in the vertical plane for constant gravity when air resistance is ignored).
• Air resistance modifies the idealized parabolic path, especially for long flights (e.g., a home run ball).
• The phrase “parabola” is tied to constant acceleration: the vertical displacement is a quadratic function of time, hence a parabolic path when projected into a vertical plane.

Center of Mass and Rigid Bodies

• Center of mass (and center of gravity, in Earth’s weak gravitational field) describes the translational motion of an extended object as if all its mass were concentrated at that point.
• For a rigid body (shape does not change), motion can be decomposed into:
  • Translational motion of the center of mass through space
  • Pure rotational motion about the center of mass
• Real-world examples: a hammer rotating about its center of mass while moving; a bicycle tire translating and rotating as it rolls.
• In the scope of these notes, we primarily analyze the translational (point-particle) motion first, then later discuss rotation.

Graphical Interpretation and Problem-Solving Strategy

• Position vs time graph:
  • Horizontal (slope 0): object is stationary.
  • Positive slope: moving in the positive direction; larger slope means faster (higher velocity).
  • Negative slope: moving in the negative direction.
  • The slope gives velocity; the curvature conveys changing velocity (acceleration).
• Velocity vs time graph:
  • Slope gives acceleration.
  • A horizontal line indicates constant velocity (zero acceleration).
• Acceleration vs time graph:
  • Horizontal line indicates constant acceleration.
  • The sign of acceleration indicates the direction of increasing velocity.
• Practical reasoning tips:
  • The tangent line on a position-time graph gives instantaneous velocity.
  • The tangent line on a velocity-time graph gives instantaneous acceleration.
  • Use signs of v and a to assess whether the object is speeding up or slowing down (speeding up when sign(v) = sign(a); slowing down when sign(v) ≠ sign(a)).
• When solving multiple-object problems, use a process of elimination on answer choices by identifying impossible qualitative features (e.g., a constant-velocity segment for a clearly curved position-time graph implies nonzero acceleration, etc.).

Worked Examples and Applications

• Example: A ball thrown upward from ground with initial speed v_0 in a vacuum (no air resistance), upward taken as positive, gravity g>0 downward (acceleration a = -g).
  • Velocity: v(t) = v_0 - g t
  • Position: y(t) = y0 + v0 t - \tfrac{1}{2} g t^2
  • Time to reach the highest point (v = 0): t{top} = \frac{v0}{g}
  • Maximum height: H = y0 + \frac{v0^2}{2 g}
  • Total flight time (up and down, symmetric in vacuum): T = \frac{2 v_0}{g}
• Example: Free fall from rest from height y_0 with downward positive coordinates:
  • Position: y(t) = y_0 - \tfrac{1}{2} g t^2
  • Velocity: v(t) = - g t (negative if upward is positive)
  • Time to hit ground (y = 0): solve 0 = y0 - \tfrac{1}{2} g t^2\Rightarrow t = \sqrt{\frac{2 y0}{g}}
• Example: Given flight time of a baseball 3.6 s (thrown upward and caught later):
  • Symmetry about the apex implies time up equals time down; thus, time to apex is t_{up} = \frac{3.6}{2} = 1.8\ ext{s}
  • From v(t) = v0 - g t, at apex v = 0, so v0 = g t_{up} = 9.81 \times 1.8 \approx 17.7 \text{ m/s}
  • If you approximate with g \approx 9.81\text{ m/s}^2 and ignore air resistance, the initial speed is about 1.8 g\;\text{m/s} as shown.
• Example: Using equation forms to derive, for vertical motion with constant acceleration, a few useful combinations:
  • \Delta y = v_0 t + \tfrac{1}{2} a t^2
  • v^2 = v0^2 + 2 a (y - y0)
  • If needed, solve for time by plugging known quantities into one of the primary equations and solving for t.

Summary: Practical Takeaways

• Three fundamental quantities in motion: displacement (Δx), velocity (v), acceleration (a).
• Displacement is the net change in position; it does not equal total distance traveled.
• Instantaneous velocity equals the slope of the position-time curve at that time; instantaneous acceleration equals the slope of the velocity-time curve at that time.
• Constant acceleration produces a parabolic position-time curve and linear velocity-time curve; the basic equations are:
  • v(t) = v_0 + a t
  • x(t) = x0 + v0 t + \tfrac{1}{2} a t^2
  • a = \frac{\Delta v}{\Delta t}
• Free fall is a canonical constant-acceleration problem with gravity near Earth's surface, typically treated with either sign convention depending on axis choice; common choice is upward positive so a = -g\,.
• The acceleration due to gravity is a constant near the surface: g \approx 9.81\ \mathrm{m/s^2}, often approximated as 10 for quick calculations.
• In real-world projectile motion, air resistance matters; neglecting it yields ideal parabolic trajectories.
• For extended bodies, translate kinetic analysis from the center of mass to the full rigid-body motion by combining translation with rotation when appropriate.
• Key qualitative reasoning tools include interpreting graphs, using tangents for instantaneous rates, and applying the sign rules for speeding up vs slowing down via the signs of velocity and acceleration.