A proton moves at 4.50 ×10^5 m / s in the horizontal...

A proton moves at $4.50 \times 10^{5} \mathrm{~m} / \mathrm{s}$ in the horizontal direction. It enters a uniform vertical electric field with a magnitude of $9.60 \times 10^{3} \mathrm{~N} / \mathrm{C}$. Ignoring any gravitational effects, find (a) the time it takes the proton to travel $5.00 \mathrm{~cm}$ horizontally, (b) its vertical displacement after it has traveled $5.00 \mathrm{~cm}$ horizontally, and (c) the horizontal and vertical components of its velocity after it has traveled $5.00 \mathrm{~cm}$ horizontally.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity or another uniform acceleration. In problems like this, the horizontal and vertical components of the motion are treated separately, with the horizontal component typically remaining at a constant velocity while the vertical component accelerates.

Kinematics in Uniform Acceleration

Kinematics in uniform acceleration involves using equations of motion which relate displacement, velocity, acceleration, and time. In this problem, the time to travel a given horizontal distance is calculated using the constant horizontal velocity, and then kinematic equations are applied to the vertical motion to determine displacement and final velocity due to constant acceleration.

Electric Force on Charged Particle

The electric force acting on a charged particle is given by F = qE, where q is the charge and E is the electric field strength. This force causes the particle to accelerate vertically according to Newton's second law (a = F/m). This is a foundational concept in electromagnetism and is used to determine the particle's vertical acceleration in the presence of an electric field.

Independence of Motion Components

The principle of independence of motion states that the horizontal and vertical motions of a particle can be analyzed separately as they do not affect each other. In this context, while the horizontal motion is unaffected by the electric field, the vertical motion is influenced solely by the acceleration due to the electric force. This separation simplifies the analysis of the two-dimensional motion in the problem.

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