acceleration

Examples of acceleration in the following topics:

• 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.

• Planetary Motion According to Kepler and Newton

  • Newton derived his theory of the acceleration of a planet from Kepler's first and second laws.
  • In addition, the magnitude of the acceleration is inversely proportional to the square of its distance from the Sun.
  • From this, Newton defined the force acting on a planet as the product of its mass and acceleration.

• Physics and Engineering: Fluid Pressure and Force

  • For fluids near the surface of the earth, the formula may be written as $p = \rho g h$, where $p$ is the pressure, $\rho$ is the density of the fluid, $g$ is the gravitational acceleration, and $h$ is the depth of the liquid in meters.

• Calculus with Parametric Curves

  • The acceleration can be written as follows with the double apostrophe signifying the second derivative:

• Vectors in Three Dimensions

  • Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors.

• 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.

• Vector-Valued Functions

  • If we differentiate a second time, we will be left with acceleration:

• Arc Length and Curvature

  • Where $\kappa$ is the curvature and $\frac{dT}{ds}$ is the acceleration vector (the rate of change of the velocity vector over time).

• Graphing on Computers and Calculators

  • c) Three-dimensional graphing: While high-end graphing calculators can graph in 3-D, GraphCalc benefits from modern computers' memory, speed, and graphics acceleration.