Download Metode Euler

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 2.00g 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 5.00cm/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…