Euler Method definition Euler method is the most

Euler Method

definition • Euler method is the most simple method that can be repaired easily. • Euler method is similar to linear regression, but in euler method the error must be included

Numerical Modeling in Particle Dynamics • the study of the dynamics of a particle focuses on describing the position, velocity, and acceleration as functions of time. • Cause-and-effect relationships exist among these quantities: Velocity causes position to change, and acceleration causes velocity to change. • Because acceleration is the direct result of applied forces, any analysis of the dynamics of a particle usually begins with an evaluation of the net force acting on the particle.

Analytical Method • we have used what is called the analytical method to investigate the position, velocity, and acceleration of a moving particle. • This method involves the identification of well-behaved functional expressions for the position of a particle, generated from algebraic manipulations or the techniques of calculus.

If a particle of mass moves under the influence of a net force ∑ F, Newton’s second law tells us that the acceleration of the particle is a= F/m In general, we apply the analytical method to a dynamics problem using the following procedure:

An Object Falling in a Vacuum— Analytical Method

The analytical method is straightforward for many physical situations • In the “real world, ” however, complications often arise that make analytical solutions difficult and perhaps beyond the mathematical abilities of most students taking introductory physics. • For example, the net force acting on a particle may depend on the particle’s position, as in cases where the gravitational acceleration varies with height • the expressions relating acceleration, velocity, position, and time are differential equations rather than algebraic ones. • Differential equations are usually solved using integral calculus and other special techniques that introductory students may not have mastered

• When such situations arise, scientists often use a procedure called numerical modeling to study motion. • In the Euler method for solving differential equations, derivatives are approximated as ratios of finite differences

Considering a small increment of time ∆t, we can approximate the relationship between a particle’s speed and the magnitude of its acceleration as • Because the acceleration is a function of time, this estimate of v(t+∆t) is accurate only if the time interval ∆t is short enough such that the change in acceleration during the interval is very small

The position x(t+∆t) of the particle at the end of the interval ∆t can be found in the same manner:

The acceleration is determined from the net force acting on the particle, and this force may depend on position, velocity, or time:

Euler method in numerical approach • …. .

One advantage of the Euler method is that the dynamics is not obscured • the fundamental relationships between acceleration and force, velocity and acceleration, and position and velocity are clearly evident. • Indeed, these relationships form the heart of the calculations. There is no need to use advanced mathematics, and the basic physics governs the dynamics.

Example A small sphere of mass 2. 00 g is released from rest in a large vessel filled with oil, where it experiences a resistive force proportional to its speed. The sphere reaches a terminal speed of 5. 00 cm/s. Using the Euler method, find the position and the acceleration of the sphere at the instant that the speed reaches 90. 0% of terminal speed.

Exercise • using Spreadsheet, solve these problems…