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Projectile motion is a form of motion experienced by an object or particle (a projectile) that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path (a trajectory) under the action of gravity only.
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
In the case of rigid bodies, the so-called quadratic velocity vector is used to describe Coriolis and centrifugal terms in the equations of motion. The name is because Q v {\displaystyle \mathbf {Q} _{v}} includes quadratic terms of velocities and it results due to partial derivatives of the kinetic energy of the body.
Consider a third-order motion-control profile with quadratic ramping and deramping phases in velocity (see figure). This picture shows a schematic diagram of jerk, acceleration, and speed, assuming all three are limited in their magnitude, when linearly going from one point to another, which are sufficiently far apart to reach the respective ...
One of the main results in the theory is that the solution is provided by the linear–quadratic regulator (LQR), a feedback controller whose equations are given below. LQR controllers possess inherent robustness with guaranteed gain and phase margin, [1] and they also are part of the solution to the LQG (linear–quadratic–Gaussian) problem.
A trivial example. In mathematics, the mountain climbing problem is a mathematical problem that considers a two-dimensional mountain range (represented as a continuous function), and asks whether it is possible for two mountain climbers starting at sea level on the left and right sides of the mountain to meet at the summit, while maintaining equal altitudes at all times.
The exact force is determined by newton's laws of motion. A solver, for example PID controllers and model predictive control, are able to bring the simulated system into a goal state. From an abstract point of view, the problem of controlling a complex physical system is a kinodynamic motion planning problem. [2]
The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Earth, the Moon, and the Sun. [2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three ...