Kinematics of Linear Motion — Comprehensive Study Notes
Coordinate Systems and Vectors
Co-ordinate systems: Cartesian (rectangular) coordinates with x- and y-axes intersecting at the origin; Points labeled as
Cartesian:
(x, y)
Polar:
(r, \theta), where r is the distance from the origin and \theta is the angle measured counterclockwise from the reference line.
Reference concepts: Origin and reference line are noted; A point’s position is described either by Cartesian coordinates (x, y) or polar coordinates (r, \theta).
Polar vs Cartesian: Cartesian uses x- and y-axes; Polar uses radius r and angle \theta relative to the reference line.
Vector Components
A vector can be described completely by its components; a component is the projection of the vector along an axis.
Rectangular components are the projections along the x- and y-axes.
x-component is the projection along the x-axis; y-component is the projection along the y-axis.
These relations assume the angle \theta is measured with respect to the x-axis. If the angle is defined differently, use the triangle directly.
Components can be positive or negative and have the same units as the original vector.
For a vector \vec{R} with magnitude R and angle \theta from the +x axis:
R_x = R \cos\theta
R_y = R \sin\theta
The magnitude and direction can be recovered from components:
R = \sqrt{Rx^2 + Ry^2}
\theta = \tan^{-1}\left(\dfrac{Ry}{Rx}\right)
Vector Addition
Adding two vectors component-wise yields the resultant components:
Rx = Ax + B_x
Ry = Ay + B_y
The resultant magnitude and direction:
R = \sqrt{Rx^2 + Ry^2}
\theta = \tan^{-1}\left(\dfrac{Ry}{Rx}\right)
Concept: When you add vectors, you add corresponding components along each axis.
Motion and Forces: Kinematics Overview
Kinematics describes motion (describing how things move) without considering forces; it is a precursor to dynamics.
Newton’s laws (described later) relate forces to motion; Newton’s principles underpin classical mechanics.
In this course, we start with kinematics: describing motion along a line (one-dimensional motion) using displacement, velocity, and acceleration.
Objects are treated as point-like in kinematics; their dimensions are neglected for motion descriptions.
Quantities can have positive or negative signs to indicate direction along the chosen axis.
Core questions as a function of time:
Where is the particle? (displacement)
How fast is it moving? (velocity)
How rapidly is it speeding up or slowing down? (acceleration)
Modelling: equations of motion describe how these quantities vary with time.
Displacement, Velocity, and Acceleration (Concepts)
Displacement (\Delta x) is the change in position; it is a vector and equals the shortest distance from initial to final position.
Velocity (v) is the rate of change of displacement; it is a vector (magnitude and direction).
Acceleration (a) is the rate of change of velocity; it is a vector.
Key relation (basic differential form):
\mathbf{v} = \dfrac{d\mathbf{x}}{dt}
\mathbf{a} = \dfrac{d\mathbf{v}}{dt}
For constant acceleration, the change in velocity over a time interval is
\Delta v = v - v_0 = a \Delta t
When motion is described along a single axis, these become scalar relationships with signs incorporated.
A standard one-dimensional setup uses the following variables:
Displacement: x (m)
Velocity: v (m/s)
Acceleration: a (m/s^2)
Relationship notes:
If velocity is positive, displacement increases in the +x direction; if negative, toward -x.
Graphical Representation of Motion
Position-time graph (x vs t):
The slope of the x-t graph gives velocity v(t).
If velocity is constant, the x-t graph is a straight line with constant slope.
If velocity is positive, x increases with time (outward); if negative, x decreases (inward).
Speed vs time graph (|v| vs t):
If speed is constant, the value on the graph is a horizontal line (constant magnitude of velocity).
Relationships:
The instantaneous velocity is the slope of the tangent to the x-t curve at a given time.
The average velocity over a time interval is the slope of the line connecting the start and end points on the x-t graph, i.e.
v_{avg} = \dfrac{\Delta x}{\Delta t}
Additional implications: the area under the v-t graph between t1 and t2 equals the displacement during that interval, and the area under the a-t graph equals the change in velocity over the interval.
Average vs Instantaneous Velocity
As you refine the time interval, the average velocity approaches the instantaneous velocity:
Instantaneous velocity is the slope of the tangent line to the x-t curve at that time.
Practical takeaway: For non-uniform motion, compute v_{avg} over small \Delta t to approximate instantaneous v.
Acceleration: Changing Velocities
Acceleration is the rate at which velocity changes with time.
If acceleration is constant, velocity changes linearly with time; the velocity-time (v-t) graph is a straight line.
The direction of acceleration relative to velocity indicates speeding up or slowing down:
If velocity and acceleration have the same direction, speed increases.
If they have opposite directions, speed decreases (decelerates).
Important nuance: An object can be accelerating even if its speed is decreasing (because velocity is a vector).
Galilean Inertia and Experimental Basis
Galileo’s experiments with balls rolling on smooth, level surfaces demonstrated inertia: a ball in motion tends to continue moving with constant velocity unless acted upon by a net external force.
This contradicted the Aristotelian view that continuous motion requires a propelling force.
The concept of inertia is a foundational principle leading to Newton’s first law of motion.
Equations of Motion (Constant Acceleration)
With constant acceleration a, at time t:
Velocity: v(t) = v_0 + a t
Position: x(t) = x0 + v0 t + \tfrac{1}{2} a t^2
These equations apply independently along each coordinate axis; vertical motion uses g = 9.8 m/s^2 (downward) for a such that a = -g when upward is positive.
Direct relation between v and x (not explicit in t):
From v = v_0 + a t, solve for t and substitute into x(t) to get the third equation:
v^2 = v0^2 + 2 a (x - x0)
Summary: These equations describe one-dimensional motion with constant acceleration and can be applied to any axis by replacing x with the appropriate coordinate (x, y, z).
Vertical motion note: gravitational acceleration is g ≈ 9.8 m/s^2 downward; if upward is chosen as positive, a = -g for free fall.
Worked Examples (One-Dimensional Motion)
Example 1: Object dropped from rest under gravity (a = -9.8 m/s^2, v0 = 0, x0 = 0)
After t = 2 s:
Velocity: v = v_0 + a t = 0 + (-9.8)(2) = -19.6 \text{ m/s}
Displacement: x = x0 + v0 t + \tfrac{1}{2} a t^2 = 0 + 0 + \tfrac{1}{2}(-9.8)(2)^2 = -19.6 \text{ m}
Interpretations: the negative sign indicates downward direction; speed ≈ 19.6 m/s; fall distance ≈ 19.6 m (≈ 64 ft).
Example 2: Gabby Thomas 200 m dash (uniformly accelerated first 7 s, constant velocity next 10 s, uniformly accelerated last 5 s; total 22 s; later 3 s to stop)
Task: sketch velocity-time and acceleration-time graphs for the entire 25 s; sketch displacement graph given distances in time slots (50 m, 100 m, 50 m, 20 m) for those intervals.
Conceptual takeaway: intentionally piecewise-accelerated motion yields a piecewise-linear v-t graph with three segments of acceleration (two accelerated phases and a constant-velocity phase).
Example 3: Two cars A and B, starting at the same position with v0 = 20 m/s east; A travels at constant speed; B decelerates with a = -2 m/s^2. When B’s speed reaches zero, compare distances traveled by A and B.
B’s stopping time: tstop = v0 / |a| = 20 / 2 = 10 s.
Distances:
xA = v0 t = 20 m/s × 10 s = 200 m
xB = v0 t - 1/2 a t^2 = 20 × 10 - 1/2 × 2 × 10^2 = 200 - 100 = 100 m
Ratio xA : xB = 200 : 100 = 2 : 1.
Example 4: A rocket traveling in space at v_0 = 1000 m/s accelerates at a = -5 m/s^2 (to stop).
Stop time: tstop = -v0 / a = 1000 / 5 = 200 s.
Distance until stop: x = v0 t + 1/2 a t^2 = 1000(200) + 1/2(-5)(200)^2 = 200000 - 100000 = 100000 m = 100 km. (Assumes x0 = 0.)
Example 5: Resultant force from two forces in different directions.
Part (a): 10 N east and 4 N west -> net 6 N east.
Part (b): 10 N east and 4 N north -> magnitude |R| = \sqrt{10^2 + 4^2} = \sqrt{116} ≈ 10.8 N; direction angle \theta = \tan^{-1}(4/10) ≈ 22^{\circ}$ north of east.
Vector components: Rx = 10 N, Ry = 4 N.
iClicker Questions (Concept Checks) and Answers
iClicker Question 1 (frictionless air-track demo): If there is a constant negative acceleration on a body with a certain velocity, what would the position-time graph look like? Choices: A, B, C, D (see slide).
iClicker Question 2: A car’s velocity-time graph shows the last six seconds before breakdown; determine acceleration from the graph. Correct answer: B.
iClicker Question 3: If you start from rest and accelerate with a given constant acceleration for a given distance, and then repeat with twice the acceleration to cover the same distance, the time required is:
Correct answer: B (time is halved).
iClicker Question 4: If an object is accelerating, then
Correct answer: C (its velocity is changing).
iClicker Question 5: If a particle has negative velocity and negative acceleration, its speed
Correct answer: B (increases) — note: speed is the magnitude; velocity and acceleration signs indicate direction; if both negative, the speed increases.
iClicker Question 6: A car goes 20 miles north at 55 mph, then returns south at 55 mph; average speed vs velocity?
Correct answer: D (The average speed is 55 mph; average velocity is 0).
iClicker Question 7: If a force of 20 N is at 30° to the horizontal, components are
Correct answer: a) X = 10 N, Y = 17.3 N (assuming standard decomposition: X = F cos 30°, Y = F sin 30°). (Note: cos 30° ≈ 0.866, sin 30° = 0.5; 20 cos 30° ≈ 17.3, 20 sin 30° = 10; thus depending on convention, option a) or c) depending on which component is labeled X vs Y. In the provided key, the choice listed as a) X=10N, Y=17.3N aligns with a 30° angle measured from the +x axis toward +y axis with F = 20 N.)
iClicker Answer Key (from slide):
1) A, 2) B, 3) B, 4) C, 5) B, 6) D, 7) B
Example 5 (continued) cross-check: The illustrated resultant magnitude and direction for (b) is approximately 10.8 N at 22° north of east.
Equations of Motion: Summary (One-Dimensional)
For constant acceleration a, at time t:
v(t) = v_0 + a t
x(t) = x0 + v0 t + \tfrac{1}{2} a t^2
Direct relation between velocity and position (eliminating t):
v^2 = v0^2 + 2 a (x - x0)
These equations apply independently to each coordinate axis; vertical motion uses g ≈ 9.8 m/s^2 with the appropriate sign convention:
If upward is positive, then for free fall a = -g.
Connections to Foundational Principles and Real-World Relevance
Kinematics sets the stage for dynamics by providing the mathematical description of motion before forces are introduced; this links directly to Newton’s laws and the study of why objects move as they do.
The concept of inertia (from Galileo) is central to understanding Newton’s first law and the baseline expectation that without external forces, motion continues uniformly.
Real-world relevance: motion under gravity, vehicle motion, sports (e.g., sprinting kinematics), projectile motion, and spaceflight dynamics all rely on these one-dimensional relationships when appropriate or on their vector extensions in multiple dimensions.
Practical considerations: idealized constant-acceleration models ignore friction, air resistance, and varying forces; in real experiments, these effects require corrections or more advanced models.
Foundational Principles and Practical Implications
One-dimensional kinematics is a stepping stone to multi-dimensional motion; all results hold on each axis separately in Cartesian coordinates.
When modeling physical systems, choose a convenient coordinate axis; decompose vectors into components to apply the one-dimensional equations component-wise.
In measurement and modeling, always check the sign conventions to ensure correct interpretation of direction and magnitude.
Ethical and practical considerations in physics experimentation include acknowledging model limitations, ensuring safe laboratory practices (e.g., frictionless track demonstrations are idealized), and appreciating the historical development of mechanics (from Aristotle to Galileo to Newton).
Quick Reference: Key Formulas and Concepts
Vector components:
Rx = R \cos\theta, \quad Ry = R \sin\theta
R = \sqrt{Rx^2 + Ry^2}, \quad \theta = \tan^{-1}\left(\dfrac{Ry}{Rx}\right)
Vector addition (component-wise):
Rx = Ax + Bx, \quad Ry = Ay + By
Displacement: \Delta x = xf - x0
Velocity and acceleration (instantaneous):
\mathbf{v} = \dfrac{d\mathbf{x}}{dt}, \quad \mathbf{a} = \dfrac{d\mathbf{v}}{dt}
Constant acceleration equations (one-dimensional):
v = v_0 + a t
x = x0 + v0 t + \tfrac{1}{2} a t^2
v^2 = v0^2 + 2 a (x - x0)
Gravity (vertical motion) convention: a = -g when upward is positive; g ≈ 9.8 m/s^2.
Graphical interpretations:
Slope of x-t is velocity; slope of v-t is acceleration; area under v-t is displacement; area under a-t is change in velocity.
Key historical note: inertia and the shift from Aristotelian to Newtonian mechanics, demonstrated by Galileo’s experiments with motion on frictionless surfaces.