A proton is fired horizontally into a 1.0×10⁵ N/C vertical electric field. It rises 1.0 cm vertically after having traveled 5.0 cm horizontally. What was the proton's initial speed?

Verified step by step guidance

1. Step 1: Identify the forces acting on the proton. The proton is subject to an electric force due to the vertical electric field. The electric force can be calculated using the formula: $F=qE$, where $q$ is the charge of the proton and $E$ is the electric field strength.
2. Step 2: Determine the acceleration of the proton due to the electric force. Using Newton's second law, $a=\frac{F}{m}$, where $m$ is the mass of the proton, substitute $F=qE$ to find the vertical acceleration.
3. Step 3: Use kinematic equations to relate the vertical displacement to the vertical acceleration. The vertical displacement $y$ is given as 1.0 cm (convert to meters: 0.01 m). Using the equation $y=\frac{1}{2}at^2$, solve for the time $t$ it takes for the proton to rise 1.0 cm.
4. Step 4: Relate the horizontal motion to the time of flight. The horizontal displacement $x$ is given as 5.0 cm (convert to meters: 0.05 m). Using the equation $x=vt$, where $v$ is the initial horizontal speed, solve for $v$ using the time $t$ found in Step 3.
5. Step 5: Combine the results from Steps 2, 3, and 4 to calculate the initial speed of the proton. Ensure all units are consistent throughout the calculations (e.g., meters, seconds, etc.).

Key Concepts

Electric Field

An electric field is a region around a charged particle where other charged particles experience a force. The strength of an electric field is measured in newtons per coulomb (N/C) and indicates how much force a unit charge would experience. In this problem, the vertical electric field exerts a force on the proton, causing it to rise as it moves horizontally.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational and other forces. In this scenario, the proton's horizontal motion is uniform while its vertical motion is influenced by the electric field. Understanding the components of projectile motion helps in analyzing the proton's trajectory and calculating its initial speed.

Kinematics Equations

Kinematics equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this problem, these equations are essential for determining the proton's initial speed based on its vertical rise and horizontal travel, allowing us to connect the forces acting on it to its motion.