definition 18.10 Velocity

open in the book · parts/03-classical-mechanics/01-kinematics.tex:253 · p. 721

ground object -- no derivation owed

Rests on

depends_on definition 7.26 Derivative of a function at a point ¶
- depends_on definition 7.16 Limit ¶
  - depends_on definition 7.2 Absolute value ¶
  - depends_on definition 7.9 Real function ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- depends_on definition 7.9 Real function ¶ ↺
depends_on definition 18.9 Position ¶
- depends_on definition 18.7 Frame of reference ¶
  - depends_on definition 18.3 Particle ¶
- depends_on definition 18.3 Particle ¶ ↺
depends_on definition 13.9 Tangent vector to a curve ¶
- depends_on definition 13.8 Curve ¶

Supports

depends_on definition 18.11 Acceleration ¶
- depends_on definition 18.27 Angular position, velocity and acceleration ¶
  - depends_on definition 18.29 Angular velocity vector ¶
    - depends_on remark 18.30 The angular velocity is an element of $\mathfrak{so}(3)$ ¶
  - depends_on phenomenon 18.31 Uniform circular motion is accelerated motion ¶
    - depends_on proposition 18.32 Tangential and centripetal components ¶
  - depends_on proposition 18.32 Tangential and centripetal components ¶ ↺
  - depends_on remark 18.28 The radian carries no dimension ¶
- depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶
- depends_on phenomenon 18.31 Uniform circular motion is accelerated motion ¶ ↺
- depends_on phenomenon 18.12 Uniformly accelerated motion ¶
  - depends_on phenomenon 20.1 phen:parabolic-trajectory ¶
    - assumes experiment Absolute Ballistic Gravimetry (1995–present) ¶
    - depends_on theorem 20.2 Parabolic trajectory ¶
      - depends_on corollary 20.3 Range, apex, time of flight ¶
      - depends_on phenomenon 20.6 Air drag ¶
      - depends_on remark 20.5 rem:projectile-idealisation ¶
    - tests experiment Galileo's Inclined Planes ¶
- depends_on proposition 18.33 Intrinsic decomposition of the acceleration ¶
  - depends_on remark 18.34 The two ways an acceleration can be zero ¶
- depends_on proposition 18.23 Transformation of velocity and acceleration ¶
  - depends_on phenomenon 18.26 Composition of velocities, and its limit ¶
depends_on definition 18.27 Angular position, velocity and acceleration ¶ ↺
depends_on definition 18.29 Angular velocity vector ¶ ↺
depends_on definition 19.29 Kinetic energy ¶
- depends_on definition 19.31 Energy ¶
  - depends_on proposition 27.20 Conservation of energy ¶
  - depends_on theorem 19.32 Conservation of energy ¶
    - depends_on phenomenon 50.6 No material surface ¶
    - depends_on phenomenon 19.33 Mechanical energy is conserved ¶
    - depends_on theorem 21.43 Conservation of energy ¶
      - depends_on proposition 29.36 Three first integrals ¶
      - depends_on theorem 27.10 Lagrange–Dirichlet ¶
- depends_on theorem 19.38 Work–energy theorem ¶
depends_on definition 19.5 Linear momentum ¶
- depends_on definition 27.2 Translational equilibrium ¶
- depends_on definition 19.18 Angular momentum ¶
  - depends_on definition 27.3 Rotational equilibrium ¶
  - depends_on definition 19.55 Angular momentum of the centre of mass ¶
    - depends_on proposition 29.14 Angular momentum and kinetic energy ¶
    - depends_on theorem 19.58 Decomposition of the angular momentum ¶
      - depends_on theorem 19.63 The spin obeys the same law about the centre of mass ¶
  - depends_on postulate 19.19 Newton's first law, rotational case ¶
    - depends_on phenomenon 19.21 Equal areas in equal times ¶
    - depends_on proposition 20.13 Eastward deflection of a body dropped from rest ¶
      - depends_on remark 20.14 Why the elementary argument gives three halves of this ¶
  - depends_on postulate 19.22 Newton's second law, rotational case ¶
    - depends_on theorem 29.25 Euler's equations ¶
      - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on proposition 29.26 The two integrals of torque-free motion ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
  - depends_on proposition 19.20 For one particle the rotational first law is not independent ¶
- depends_on definition 19.51 Linear momentum of the centre of mass ¶
  - depends_on definition 19.55 Angular momentum of the centre of mass ¶ ↺
- depends_on postulate 19.6 Newton's first law ¶
  - depends_on phenomenon 19.7 Persistence of uniform motion ¶
  - depends_on postulate 19.19 Newton's first law, rotational case ¶ ↺
  - depends_on postulate 19.8 Newton's second law ¶
    - depends_on definition 19.30 Conservative force and potential energy ¶
      - depends_on definition 27.13 Central force ¶
      - depends_on definition 19.31 Energy ¶ ↺
      - depends_on proposition 19.71 The Newtonian closed system is a Lagrangian system ¶
    - depends_on definition 19.34 Work ¶
      - depends_on definition 19.35 Power ¶
      - depends_on proposition 19.36 Work of a torque ¶
      - depends_on theorem 19.38 Work–energy theorem ¶ ↺
    - depends_on phenomenon 49.16 Pulsars: clocks at nuclear density ¶
    - depends_on phenomenon 42.1 Universality of free fall ¶
      - depends_on phenomenon 53.5 Universality of free fall, as measured ¶
      - depends_on phenomenon 42.4 The lunar orbit is not polarized toward the Sun ¶
    - depends_on phenomenon 19.10 Force, mass and acceleration ¶
    - depends_on postulate 19.22 Newton's second law, rotational case ¶ ↺
    - depends_on postulate 19.48 Newton's third law ¶
      - depends_on phenomenon 19.50 Recoil and the conservation of momentum ¶
      - depends_on postulate 19.49 Net internal force ¶
      - depends_on postulate 19.61 Newton's third law, strong form ¶
      - depends_on theorem 27.30 Reduction of the two-body problem ¶
      - depends_on theorem 19.52 The centre of mass moves as a particle ¶
    - depends_on theorem 20.2 Parabolic trajectory ¶ ↺
    - depends_on theorem 29.51 Equation of motion in a rotating frame ¶
      - depends_on phenomenon 42.4 The lunar orbit is not polarized toward the Sun ¶ ↺
  - depends_on proposition 19.20 For one particle the rotational first law is not independent ¶ ↺
- depends_on postulate 19.8 Newton's second law ¶ ↺
depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶ ↺
depends_on proposition 18.23 Transformation of velocity and acceleration ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Derivative of a function at a point	declared	parts/03-classical-mechanics/01-kinematics.tex:261
`depends_on`	→	Position	declared	parts/03-classical-mechanics/01-kinematics.tex:261
`depends_on`	→	Tangent vector to a curve	declared	parts/03-classical-mechanics/01-kinematics.tex:261
`depends_on`	←	Acceleration	declared	parts/03-classical-mechanics/01-kinematics.tex:283
`depends_on`	←	Angular position, velocity and acceleration	declared	parts/03-classical-mechanics/01-kinematics.tex:846
`depends_on`	←	Angular velocity vector	declared	parts/03-classical-mechanics/01-kinematics.tex:879
`depends_on`	←	Kinetic energy	declared	parts/03-classical-mechanics/02-newtonian-dynamics.tex:815
`depends_on`	←	Linear momentum	declared	parts/03-classical-mechanics/02-newtonian-dynamics.tex:99
`depends_on`	←	Velocity and acceleration in plane polar coordinates	declared	parts/03-classical-mechanics/10-central-forces-statics.tex:419
`depends_on`	←	Transformation of velocity and acceleration	declared	parts/03-classical-mechanics/01-kinematics.tex:709