Essential Kinematics Formulas and Motion Principles

Position Vector and Components

The position vector r describes an object's location in space. Its components can be expressed in Cartesian or polar coordinates:

• Position Vector: r = xi + yj
• Cartesian X-component: x = r cos θ
• Cartesian Y-component: y = r sin θ
• Magnitude of Position Vector: r = √(x2 + y2)
• Angle of Position Vector: tan θ = y / x

Displacement

Displacement (Δr) is the change in an object's position:

• Final Displacement: Δr = rfinal - rinitial

Speed and Velocity

Speed is the magnitude of velocity. Velocity is a vector quantity describing the rate of change of position:

• Average Speed: vavg = Δr / Δt
• Instantaneous Speed: v = |dr / dt|
• Average Velocity: vavg = Δr / Δt
• Instantaneous Velocity: v = dr / dt

Acceleration

Acceleration (a) is the rate of change of velocity:

• Average Acceleration: aavg = Δv / Δt
• Instantaneous Acceleration: a = dv / dt

Uniform Rectilinear Motion (URM)

Uniform Rectilinear Motion describes motion in a straight line with constant velocity (zero acceleration). The following equations apply:

• Velocity: v = Δx / Δt
• Mean Velocity: vmean = (v0 + v) / 2
• Final Velocity: v = v0 + at
• Position (Constant Velocity): x = x0 + vt
• Position (Constant Acceleration): x = x0 + v0t + (1/2)at2
• Velocity-Displacement Relation: v2 - v02 = 2aΔx

Vertical Motion: Free Fall & Projectile Launch

These equations describe motion under the influence of gravity, assuming constant gravitational acceleration (g).

Free Fall (Downward Motion)

For an object dropped from rest, assuming downward is the positive direction:

• Velocity Equation: v = gt
• Position Equation (from rest, y₀ = 0): y = (1/2)gt2

Vertical Launch (Upward Motion)

For an object launched vertically upwards, assuming upward is the positive direction:

• Velocity Equation: v = v0 - gt
• Position Equation: y = y0 + v0t - (1/2)gt2

Key Parameters for Vertical Launch

• Maximum Height: At the maximum height, the vertical velocity of the body is zero (v = 0).
• Time to Max Height: t = v0 / g
• Maximum Height Reached: ymax = v02 / (2g)
• Time of Flight: The time taken for the body to return to its initial height (y = 0).
• Total Time of Flight: tflight = 2v0 / g

Projectile Motion: Horizontal Launch

Describes the motion of an object launched horizontally, where the initial vertical velocity is zero.

Horizontal Component (Uniform Motion)

• Horizontal Position: x = v0xt (where v0x is the initial horizontal velocity)
• Horizontal Velocity: vx = v0x (constant)

Vertical Component (Free Fall)

Assuming upward is positive and y₀ is the initial height:

• Vertical Position: y = y0 - (1/2)gt2
• Vertical Velocity: vy = -gt

Resultant Velocity and Position

• Resultant Velocity Magnitude: v = √(vx2 + vy2)
• Position Vector: r = (v0xt)i + (y0 - (1/2)gt2)j
• Velocity Vector: v = v0xi - gtj

General Projectile Motion (Parabolic)

Describes the motion of an object launched at an angle to the horizontal.

Initial Velocity Components

If an object is launched with initial speed v0 at an angle θ above the horizontal:

• Horizontal Component: v0x = v0 cos θ
• Vertical Component: v0y = v0 sin θ

Motion Components

• Horizontal Position: x = v0xt
• Vertical Position: y = y0 + v0yt - (1/2)gt2
• Horizontal Velocity: vx = v0x (constant)
• Vertical Velocity: vy = v0y - gt

Resultant Vectors

• Position Vector: r = xi + yj
• Velocity Vector: v = vxi + vyj

Maximum Range

The maximum horizontal distance covered when the object returns to its initial height (y = 0).

• Time of Flight: t = 2v0y / g = 2v0 sin θ / g
• Maximum Range: xmax = v02 sin(2θ) / g

Maximum Height

The highest vertical point reached during parabolic motion. At this point, the vertical velocity is zero (vy = 0).

• Time to Max Height: t = v0y / g = v0 sin θ / g
• Maximum Height: ymax = v02 sin2θ / (2g)

Superposition of Uniform Motions

When an object undergoes multiple independent uniform motions simultaneously, their effects can be combined using vector addition:

• Displacement: Δr = Δxi + Δyj
• Velocity: v = vxi + vyj
• Position Components: x = vxt, y = vyt

Uniform Circular Motion (UCM)

Describes motion in a circular path at a constant speed.

Key Definitions

• Angular Velocity (ω): ω = Δθ / Δt
• Tangential Speed (v): v = Δs / Δt (where Δs is arc length)
• Angular Position: θ = θ0 ± ωt

Period and Frequency

• Period (T): Time for one complete revolution. T = t / N (where N is number of revolutions)
• Frequency (f): Number of revolutions per unit time. f = N / t
• Relationship: f = 1 / T

Angular Velocity and Centripetal Acceleration

• Angular Velocity: ω = 2π / T
• Angular Velocity: ω = 2πf
• Centripetal Acceleration (ac): ac = ω2r
• Centripetal Acceleration: ac = v2 / r
• Centripetal Acceleration: ac = (2π / T)2r

Uniformly Accelerated Circular Motion (UACM)

Describes motion in a circular path with constant angular acceleration.

• Angular Acceleration (α): α = Δω / Δt
• Angular Velocity Equation: ω = ω0 + αt
• Angular Position Equation: θ = θ0 + ω0t + (1/2)αt2