Notes on Motion in a Plane (Chapter 3)

3.1 Introduction

• Review of straight-line motion concepts from the previous chapter: position, displacement, velocity, acceleration.

• In one dimension, directions are handled by signs (+, −).

• For motion in a plane (2D) or space (3D), use vectors to describe position, displacement, velocity, and acceleration.

• Goals: learn the language of vectors, how to add/subtract/multiply vectors, and the effect of multiplying by a real number.

• Applications: define velocity and acceleration in a plane, study motion in a plane with constant acceleration, projectile motion, and uniform circular motion.

• Equations developed in this chapter extend to three dimensions.

3.2 Scalars and vectors

• Scalars: quantities with magnitude only; specified by a number with a unit.

  • Examples: distance, mass, temperature, time.

  • Scalar algebra: ordinary arithmetic rules apply.

• Vectors: quantities with both magnitude and direction; obey triangle law (or parallelogram law) of addition.

  • Represented by boldface letters (e.g., v). When handwritten, an arrow over the letter denotes a vector (e.g.,  v ).

  • Magnitude often denoted |v| = v.

  • Examples of vector quantities: displacement, velocity, acceleration, force.

3.2.1 Position and Displacement Vectors

• Choose origin O and consider the position of an object at times t and t′: P and P′.

• Position vector from O to P: r = OP; to P′: r′ = OP′.

• The displacement vector from P to P′ is PP′, with tail at P and head at P′.

• Important: Displacement is the straight line between initial and final positions and is independent of the actual path taken.

• Example: Different paths between P and Q (PABCQ, PDQ, PBEFQ) have the same displacement vector PQ.

3.2.2 Equality of Vectors

• Two vectors A and B are equal iff they have the same magnitude and direction: A = B.

• Equality can be checked by translating B parallel to itself until tails coincide; if heads also coincide, they are equal.

• Vectors can have the same magnitude but different directions (not equal).

• Free vectors vs localized vectors:

  • Free vectors: location/line of action does not matter (translation leaves the vector unchanged).

  • Localized vectors: location of the vector is important in some applications.

3.3 Multiplication of vectors by real numbers

• Scalar multiplication: multiplying a vector A by a positive scalar λ yields a vector with magnitude |λ||A| in the same direction as A.

  • |λA| = λ|A| for λ > 0.

• If λ is negative, the resulting vector points in the opposite direction and has magnitude |λ|·|A|.

• The scalar multiplier may carry its own dimensions; then the dimensions of λA are the product of the dimensions of λ and A.

  • Example: multiplying a constant velocity vector by time gives a displacement vector.

3.4 Addition and subtraction of vectors — graphical method

• Vectors obey the triangle (parallelogram) law of addition.

• Graphical (head-to-tail) addition:

  • place the tail of B at the head of A; the line from the tail of A to the head of B is the resultant R = A + B.

• Triangle method is equivalent to the parallelogram method.

• Commutativity: A + B = B + A. (3.1)

• Associativity: (A + B) + C = A + (B + C). (3.2)

• Null vector: A + (−A) = 0; magnitude |0| = 0. (3.3)

• The zero vector properties: A + 0 = A; 0·A = 0; 0·(A) = 0.

• Subtraction: A − B = A + (−B). (3.5)

• Parallelogram method and triangle method yield the same resultant.

3.5 Resolution of vectors

• Any vector A can be expressed as a linear combination of two non-colinear vectors a and b: A = λa + μb (λ, μ ∈ ℝ).

• Construction: through O draw a line parallel to a, through P draw a line parallel to b; their intersection Q gives the decomposition.

• If using unit vectors, we can resolve along unit directions; unit vectors along x, y, z are denoted as  i ,  j ,  k  with |i| = |j| = |k| = 1 and mutually perpendicular.

• In two dimensions, resolve along i-hat and j-hat:

  • A = Ax  i  + Ay  j ,

  • Ax = A cos θ, Ay = A sin θ, where θ is the angle with the x-axis. (3.13)

• General component forms in 3D: for α, β, γ as the angles with the x-, y-, z-axes, respectively,

  • Ax = |A| cos α, Ay = |A| cos β, A_z = |A| cos γ. (3.16a)

  • |A|^2 = Ax^2 + Ay^2 + A_z^2. (3.16c)

• Position vector in components: r = x i  + y j  + z k . (3.17)

• For a vector in the x–y plane, A = Ax  i  + Ay  j  with Ax = A cos θ, Ay = A sin θ. (3.12, 3.13)

3.6 Vector addition – analytical method

• When adding vectors, it can be done by adding corresponding components:

  • Suppose A = Ax  i  + Ay  j  and B = Bx  i  + By  j ,

  • Then R = A + B has components Rx = Ax + Bx and Ry = Ay + By. (3.18)

• This analytical method extends to three dimensions similarly with k-hat: Rijk = (Ax + Bx)  i  + (Ay + By)  j  + (Az + B_z)  k . (3.22)

• For two vectors in components, a compound expression for the resultant is: R = (Ax + Bx) i  + (Ay + By) j . (3.21)

• The Laws of Cosines and Sines connect magnitudes and angles when using the analytical method (see 3.24a–3.24f in examples).

3.7 Motion in a plane

• The plane motion can be described with vectors in two perpendicular directions.

• Position vector in a plane is r = x i  + y j .

• Displacement: Δr = r′ − r = Δx  i  + Δy  j . (3.26)

• Average velocity:

  • v̄ = Δr / Δt, i.e., components vx = Δx/Δt, vy = Δy/Δt. (3.27)

• Instantaneous velocity: limit of average velocity as Δt → 0:

  • v = dr/dt = vx  i  + vy  j ; also, |v| = sqrt(vx^2 + vy^2). (3.28, 3.30)

• Velocity direction is tangential to the path.

• Acceleration: average acceleration and instantaneous form:

  • a = dv/dt; components a = ax  i  + ay  j , with |a| = sqrt(ax^2 + ay^2). (3.31, 3.32)

• In 2D with constant acceleration, the position and velocity evolve as:

  • v(t) = v₀ + at, and

  • displacement over time t is given by r = r₀ + v₀t + (1/2)at^2. (3.33)

• The motion can be viewed as the superposition of two independent 1D motions along perpendicular axes.

3.7.1 Position vector and displacement (continuation)

• For a trajectory r(t) = x(t)  i  + y(t)  j :

  • Displacement: Δr = r′ − r = (Δx, Δy). (3.26)

  • Average velocity: v̄ = (Δx/Δt, Δy/Δt).

  • Instantaneous velocity components: vx = dx/dt, vy = dy/dt; magnitude v = sqrt(vx^2 + vy^2); direction tan θ = vy/vx.

3.8 Motion in a plane with constant acceleration

• If acceleration a is constant, with initial velocity v₀ at t = 0, velocity at time t is

  • v(t) = v₀ + at. (3.33a)

• Position evolves as

  • r(t) = r₀ + v₀t + (1/2)at^2. (3.33a)

• In component form: vx = v₀x + ax t, vy = v₀y + ay t; x(t) and y(t) follow accordingly. (3.33b)

• Example demonstrates how to solve kinematics in two dimensions with constant acceleration by integrating components separately.

3.9 Projectile motion

• Projectile motion treats horizontal and vertical motions separately:

  • ax = 0, ay = −g (gravity downward). (3.35)

  • Initial velocity components: v₀x = v₀ cos θ₀, v₀y = v₀ sin θ₀. (3.36)

• Position equations (with origin at launch):

  • x(t) = v₀x t = (v₀ cos θ₀) t,

  • y(t) = v₀y t − (1/2) g t^2 = (v₀ sin θ₀) t − (1/2) g t^2. (3.34b, 3.37)

• Velocity components: vx(t) = v₀x, vy(t) = v₀y − g t. (3.38)

• Path equation (eliminating t): y = x tan θ₀ − (g x^2)/(2 v₀^2 cos^2 θ₀). (3.39)

• Key results:

  • The trajectory is a parabola (assuming air resistance negligible). (3.39)

  • Time of flight: Tf = 2 v₀ sin θ₀ / g. (3.40b)

  • Time to maximum height: t_m = v₀ sin θ₀ / g. (3.40a)

  • Maximum height: h = (v₀^2 sin^2 θ₀)/(2g). (3.41)

  • Horizontal range: R = (v₀^2 sin 2θ₀)/g. (3.42a)

  • For maximum range, θ₀ = 45°; R_max = v₀^2/g. (3.42b)

• Examples illustrate horizontal launch, projectile with various angles, and comparisons like Galileo’s range symmetry around 45°.

3.10 Uniform circular motion

• Definition: motion along a circle with constant speed v is uniform circular motion; velocity is always tangent to the path.

• Centripetal acceleration is directed toward the center of the circle.

• Derivation (sketch): for small time Δt, Δv is directed toward the center; ac = Δv/Δt toward center. In the limit Δt → 0, ac = v^2/R. (3.43)

• Angular speed ω is the rate of change of angular displacement: ω = Δθ/Δt. (3.44)

• Relations:

  • v = Rω, and a_c = Rω^2. (3.45, 3.46)

  • Time period T and frequency ν: ω = 2πν, v = 2πRν, a_c = 4π^2 ν^2 R. (3.46, 3.47, 3.48)

• Example: an insect in a circular groove (R = 12 cm) completes 7 revolutions in 100 s:

  • ω = 2π/T = 2π(7/100) = 0.44 rad/s,

  • v = ωR = 0.44 × 12 cm ≈ 5.3 cm/s,

  • a_c = v^2/R ≈ (0.44)^2 × 12 cm ≈ 2.3 cm/s^2.

  • Note: direction of a_c is toward the center and not a constant vector due to changing direction. (Example 3.9)

SUMMARY

• Scalars vs vectors: scalars have magnitude only; vectors have magnitude and direction.

• Vector operations:

  • Scalar multiplication scales magnitude and may reverse direction if negative.

  • Vector addition obeys triangle/parallelogram laws; commutative and associative.

  • Zero vector 0 has magnitude 0; A + 0 = A; 0·A = 0; A − B = A + (−B).

• Resolution and components:

  • Any vector can be resolved into components along chosen directions using unit vectors (i, j, k).

  • In 2D, A = Ax i + Ay j with Ax = A cos θ, Ay = A sin θ.

• Vector addition (analytical): Rx = Ax + Bx, Ry = Ay + By; R = √(Rx^2 + Ry^2).

• Motion in a plane:

  • Displacement Δr = r′ − r; velocity is tangent to the path; acceleration is the rate of change of velocity.

  • Instantaneous velocity: v = dr/dt; instantaneous acceleration: a = dv/dt.

  • In constant acceleration, r and v follow compact component forms: r(t) = r₀ + v₀ t + (1/2) a t^2; v(t) = v₀ + a t.

• Projectile motion:

  • Independence of horizontal and vertical motions under constant gravity (Galileo’s principle).

  • Parabolic path; key formulas for x(t), y(t), vx(t), vy(t); time of flight Tf; maximum height h; horizontal range R.

• Uniform circular motion:

  • Constant speed, acceleration toward the center with magnitude ac = v^2/R or ac = ω^2 R.

  • Relationship between linear and angular quantities: v = Rω, ω = 2πν; a_c = 4π^2 ν^2 R.

• Common conceptual notes:

  • Displacement depends only on endpoints, path length depends on the path taken.

  • Average speed vs magnitude of average velocity.

  • Projection of motion in plane can be analyzed as two independent 1D motions.

  • Trajectory shape depends on initial conditions in addition to acceleration.

POINTS TO PONDER

• Path length vs displacement magnitude; conditions for equality.

• Whether average speed is always greater than or equal to the magnitude of average velocity.

• The vector equations (3.33a) and (3.34a) hold independent of axis choice.

• Kinematic equations for uniform acceleration do not apply to uniform circular motion because magnitude is constant but direction changes.

• Resultant velocity of two velocities v1 and v2 is v = v1 + v2; distinguish from relative velocity v12 = v1 − v2.

• In circular motion, resultant acceleration is toward the centre only if speed is constant.

• Trajectory shape depends on initial conditions in addition to acceleration (gravity can yield straight-line or parabolic paths depending on initial velocity and position).

EXERCISES (selected topics and structure)

• 3.1 Determine whether quantities are scalars or vectors (volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity).

• 3.2 Identify the two scalar quantities in a list of items (e.g., force, angular momentum, work, current, etc.).

• 3.3 Identify the only vector quantity in a list (e.g., Temperature, pressure, impulse, time, etc.).

• 3.4 Assess which algebraic operations are meaningful for scalars/vectors (e.g., adding scalars; multiplying a vector by a scalar; etc.).

• 3.5 True/false statements about vector magnitude, components, path length vs displacement, average speed vs average velocity, and null vectors.

• 3.6 Prove vector inequalities: |a+b| ≤ |a| + |b|, |a+b| ≥ ||a| − |b||, etc.; when equality holds.

• 3.7 If a + b + c + d = 0, evaluate several implications for magnitudes and planar relationships.

• 3.8 Three girls travel from P to Q on a circle; compare displacement magnitudes to actual path lengths for different routes.

• 3.9 A cyclist’s trip in a circular park: determine net displacement, average velocity, and average speed.

• 3.10 A motorist follows left-hand turns; determine displacements after several turns and compare to total path length.

• 3.11 A passenger’s taxi route: compute average speed vs average velocity magnitude.

• 3.12 Ceiling problem: maximum horizontal distance a ball can travel without hitting the ceiling.

• 3.13 Cricket throw: given horizontal range, determine maximum height possible.

• 3.14 Circular motion: a stone on a string whirled in a circle; compute centripetal acceleration magnitude and direction.

• 3.15 Aircraft horizontal loop: compare centripetal acceleration with gravity.

• 3.16 True/false statements about net acceleration in circular motion, velocity direction, and average acceleration over a cycle.

• 3.17-3.19 Vector practice problems involving components, unit vectors, and projections.

• 3.20 True statements about general motion relations (averages, position/velocity relations, kinematic equations).

• 3.21 True/false about scalar quantities’ properties and observer orientation.

• 3.22 An aircraft at height: what is the speed given angular change with respect to a ground point?

This set of notes captures the major and minor points from the transcript, including key definitions, laws, formulas, and example problems, and presents them in a structured, study-ready format with explicit LaTeX for all mathematical expressions.