Equation of Motion in a Uniform Electric Field

This page derives the equations of motion for a charged particle in a uniform electric field. The approach is similar to that used for the gravitational field, but with the electric force replacing gravity.

Definition: A uniform electric field is one where the electric field strength E is constant in magnitude and direction throughout the region of interest.

The derivation begins with Newton's second law in a Galilean reference frame, F = ma. For a particle with charge q in an electric field E, the force is given by Fe = qE.

Vocabulary: Electric field - A region in space where an electric charge experiences a force.

Equating the electric force to mass times acceleration yields:

a = qE/m

This acceleration is constant, similar to the gravitational case, but its magnitude depends on the charge-to-mass ratio of the particle.

The page then presents the initial conditions and integrates the acceleration equation twice to obtain the position equations:

x = V₀t y = ½ × $qE/m$ × t²

Example: These equations describe the motion of a charged particle initially moving horizontally in a vertical electric field.

The trajectory equation is derived by eliminating t:

y = $qE / (2mV₀²)$ × x²

Highlight: This parabolic equation is crucial for solving Particule chargée dans un champ électrique uniforme exercices corrigés pdf problems.

The page concludes with a summary of the equations of motion and trajectory for a charged particle in a uniform electric field, providing a comprehensive set of tools for analyzing such systems.

Equation of Motion in a Uniform Gravitational Field

This page focuses on deriving the equations of motion for an object in a uniform gravitational field. The derivation starts with Newton's second law of motion and applies it to the specific case of an object in a gravitational field.

Definition: A uniform gravitational field is one where the gravitational acceleration g is constant throughout the region of interest.

The page begins by stating the fundamental equation F = ma in a Galilean reference frame. It then identifies that in a uniform gravitational field, the force is given by P = mg, where m is the mass and g is the gravitational acceleration vector.

Vocabulary: Galilean reference frame - A frame of reference in which Newton's laws of motion are valid.

The derivation proceeds by equating the gravitational force to mass times acceleration, resulting in g = a. This is then separated into components, with the vertical acceleration being -g.

Initial conditions are introduced, and the equation is integrated twice to obtain the position equations. The resulting equations are:

x = cos(α) × V₀ × t y = -½ × g × t² + sin(α) × V₀ × t

Example: These equations describe the motion of a projectile launched at an angle α with initial velocity V₀.

The page concludes by presenting the trajectory equation, which relates y to x:

y = -½ × g × $x / (cos(α) × V₀)$² + tan(α) × x

Highlight: This equation is particularly useful for analyzing the path of projectiles and solving Exercice corrigé mouvement d'un projectile problems.