Scalars and vectors
A scalar has only magnitude: mass, time, distance, speed, energy. A vector has magnitude and direction and adds by the triangle law: displacement, velocity, acceleration, force.
A vector is written in bold, A, or with an arrow; its magnitude is |A| or A.
Two vectors are equal if they have the same magnitude and direction.
Multiplying a vector by a positive number λ changes its magnitude by λ and keeps its direction; multiplying by −1 reverses it.
Adding vectors
Triangle law: place the tail of B at the head of A. The resultant R = A + B runs from the tail of A to the head of B. The parallelogram law gives the same result.
For two vectors of magnitudes A and B at angle θ:
R = √(A² + B² + 2AB cos θ), and the angle α the resultant makes with A satisfies tan α = B sin θ ⁄ (A + B cos θ).
θ = 0°: R = A + B (maximum).
θ = 180°: R = |A − B| (minimum).
θ = 90°: R = √(A² + B²).
Vector addition is commutative (A + B = B + A) and associative. Subtracting B means adding −B.
Unit vectors and components
A unit vector has magnitude 1 and shows only direction. î, ĵ, k̂ point along the x, y and z axes.
Any vector in the xy-plane can be resolved into components:
A = Aₓ î + A_y ĵ, with Aₓ = A cos θ and A_y = A sin θ, where θ is the angle with the x-axis.
Conversely, A = √(Aₓ² + A_y²) and tan θ = A_y ⁄ Aₓ.
Adding vectors is easiest by components: if R = A + B, then Rₓ = Aₓ + Bₓ and R_y = A_y + B_y.
Example. A force of 10 N acts at 30° to the x-axis. Its components are 10 cos 30° ≈ 8.66 N and 10 sin 30° = 5 N.
Motion in two dimensions
Position r = x î + y ĵ. Velocity v = dr/dt and acceleration a = dv/dt. For constant acceleration, the equations of motion hold separately for each component:
v = v₀ + at
r = r₀ + v₀t + ½at²
The x- and y-motions are independent of each other, which is the key to projectile motion.
Relative velocity
The velocity of A relative to B is v_AB = v_A − v_B.
Example: rain and an umbrella. Rain falls vertically at 30 m/s. A cyclist rides east at 10 m/s. Relative to the cyclist, the rain has a horizontal component of 10 m/s westward and a vertical component of 30 m/s downward. So it appears to come at angle θ to the vertical with tan θ = 10/30, about 18°. The cyclist should tilt the umbrella forward by that angle.
Projectile motion
An object thrown into the air and moving only under gravity is a projectile. Ignoring air resistance:
the horizontal velocity stays constant (no horizontal acceleration),
the vertical motion has constant acceleration g downward.
For a projectile launched with speed u at angle θ above the horizontal:
Quantity | Formula |
|---|---|
Equation of path | y = x tan θ − g x² ⁄ (2u² cos² θ), a parabola |
Time of flight | T = 2u sin θ ⁄ g |
Maximum height | H = u² sin² θ ⁄ (2g) |
Horizontal range | R = u² sin 2θ ⁄ g |
The range is maximum at θ = 45°, where R_max = u²/g. Angles θ and (90° − θ) give the same range: 30° and 60° land at the same point, but the 60° projectile goes higher and stays in the air longer.
Example. A ball is kicked at 20 m/s at 30° (g = 10 m/s²).
T = 2 × 20 × 0.5 ⁄ 10 = 2 s
H = 400 × 0.25 ⁄ 20 = 5 m
R = 400 × sin 60° ⁄ 10 ≈ 34.6 m
Uniform circular motion
An object moving in a circle at constant speed is in uniform circular motion. Its velocity keeps changing direction, so it is accelerating.
Angular speed ω = Δθ/Δt (rad/s); the speed v = ωr.
The centripetal acceleration points towards the centre: a_c = v²/r = ω²r.
With time period T and frequency ν: ω = 2π/T = 2πν.
Example. A stone tied to a 0.5 m string makes 2 revolutions per second. ω = 2π × 2 = 4π rad/s, v = ωr = 2π ≈ 6.28 m/s, and a_c = ω²r = 16π² × 0.5 ≈ 79 m/s².
Summary
Vectors add by the triangle or parallelogram law, or by components.
Horizontal and vertical motions of a projectile are independent; the path is a parabola.
T = 2u sin θ/g, H = u² sin² θ/2g, R = u² sin 2θ/g; R is greatest at 45°.
In uniform circular motion, a_c = v²/r = ω²r, directed towards the centre.