velocity

Examples of velocity in the following topics:

• Tangent and Velocity Problems

  • Velocity is defined as rate of change of displacement.
  • The average velocity becomes instantaneous velocity at time t.
  • Instantaneous velocity is always tangential to trajectory.
  • Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.
  • Its slope is the velocity at that point.

• Calculus with Parametric Curves

  • Thus, one can describe the velocity of a particle following such a parametrized path as:
  • where $v$ is the velocity, $r$ is the distance, and $x$, $y$, and $z$ are the coordinates.
  • The horizontal velocity is the time rate of change of the $x$ value, and the vertical velocity is the time rate of change of the $y$ value.

• Work

  • For moving objects, the rate of the work done by a force (measured in joules/second, or watts) is the scalar product of the force (a vector) and the velocity vector of the point of application.
  • This scalar product of force and velocity is classified as instantaneous power delivered by the force.
  • Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.
  • As the point moves, it follows a curve $X$, with a velocity $v$, at each instant.

• Directional Derivatives and the Gradient Vector

  • The directional derivative represents the instantaneous rate of change of the function, moving through $\mathbf{x}$ with a velocity specified by $\mathbf{v}$.
  • The directional derivative of a multivariate differentiable function along a given vector $\mathbf{v}$ at a given point $\mathbf{x}$ intuitively represents the instantaneous rate of change of the function, moving through $\mathbf{x}$ with a velocity specified by $\mathbf{v}$.

• Vector Fields

  • Vector fields can be thought to represent the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (the rate of change of volume of a flow) and curl (the rotation of a flow).
  • In the case of the velocity field of a moving fluid, a velocity vector is associated to each point in the fluid.

• Higher Derivatives

  • The second derivative of $x$ is the derivative of $x'(t)$, the velocity, and by definition is the object's acceleration.
  • Acceleration is the time-rate of change of velocity, and the second-order rate of change of position.

• Differentiation and Rates of Change in the Natural and Social Sciences

  • For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration.

• Arc Length and Speed

  • Since length is a magnitude that involves position, it is easy to deduce that the derivative of a length, or position, will give you the velocity—also known as speed—of a function.
  • Velocity is the rate of change of a position with respect to time.

• Arc Length and Curvature

  • In order to find the value of the curvature, we need to take the parameter time, s, and the unit tangent vector, which in this case is the same as the unit velocity vector, T, which is also a function of time.The curvature is a magnitude of the rate of change of the tangent vector, T:
  • Where $\kappa$ is the curvature and $\frac{dT}{ds}$ is the acceleration vector (the rate of change of the velocity vector over time).

• Applications of Second-Order Differential Equations

  • $\omega_0$ is called angular velocity, and the constants $A$ and $\phi$ are determined from initial conditions of the motion.
  • In many vibrating systems the frictional force $Ff$ can be modeled as being proportional to the velocity v of the object: $Ff = −cv$, where $c$ is called the viscous damping coefficient.