Problem 45 - College Physics

Question

An electron is ejected horizontally at a speed of \(1.5 \times 10^{6} \mathrm{~m} / \mathrm{s}\) from the electron gun of a computer monitor. If the viewing screen is \(35 \mathrm{~cm}\) from the end of the gun, how far will the electron travel in the vertical direction before hitting the screen? Based on your answer, do you think designers need to worry about this gravitational effect?

Step-by-Step Solution

Answer

The electron falls approximately \(2.66 \times 10^{-14} \text{ m}\) vertically; this gravitational effect is negligible.

Step 1: Identify Horizontal and Vertical Components

The horizontal component of the electron's velocity is given as \(v_{x} = 1.5 \times 10^6 \text{ m/s}\). The vertical component of velocity due to gravity will initially be zero because it is ejected horizontally.

Step 2: Calculate Time of Travel

Use the horizontal distance \(d = 35 \text{ cm} = 0.35 \text{ m}\) to find the time \(t\) it takes for the electron to reach the screen. Use the formula \(d = v_{x} \cdot t\). Solving for \(t\), we get:\[t = \frac{d}{v_{x}} = \frac{0.35}{1.5 \times 10^6} \approx 2.33 \times 10^{-7} \text{ s}\]

Step 3: Calculate Vertical Displacement

Using the time we calculated, determine the vertical displacement \(y\) due to gravity. Use the formula for vertical motion \(y = \frac{1}{2} g t^2\), where \(g = 9.8 \text{ m/s}^2\): \[y = \frac{1}{2} \cdot 9.8 \cdot (2.33 \times 10^{-7})^2 \approx 2.66 \times 10^{-14} \text{ m}\]

Step 4: Conclusion of the Gravitational Effect

Since the calculated vertical displacement \(y \approx 2.66 \times 10^{-14} \text{ m}\) is extremely small relative to the screen distance and the dimensions of the screen, designers do not need to worry about this gravitational effect on the electron's path.

Key Concepts

Horizontal Velocity

In projectile motion, horizontal velocity is a crucial factor when dealing with motion where an object is projected parallel to the ground. For the electron in question, ejected horizontally from the electron gun of a computer monitor, the horizontal velocity is given. It's valued at \(1.5 \times 10^6 \mathrm{~m/s}\). This velocity is constant throughout the motion since there are no horizontal forces acting on the electron.

• It defines how fast the electron moves horizontally across the screen's plane.
• It's measured in meters per second \((m/s)\).

In cases like these, the horizontal velocity is unaffected by gravity, which only influences the vertical motion. Therefore, you can use this velocity to calculate the time it takes for the electron to travel a known horizontal distance, which further helps in analyzing other aspects of projectile motion.

Vertical Displacement

Vertical displacement is the measure of how much an object moves in the vertical direction during its flight. In this scenario, the electron is initially ejected along a horizontal path, meaning its initial vertical velocity is zero.

• Gravity causes a downward acceleration that affects the vertical direction.
• The vertical displacement can be calculated using the formula \(y = \frac{1}{2} g t^2\), where \(g\) is the acceleration due to gravity \((9.8 \mathrm{~m/s}^2)\).

The vertical displacement helps determine how far the electron moves vertically by the time it hits the viewing screen. Even though it is shown to be very small (about \(2.66 \times 10^{-14} \text{ m}\) in this exercise), it crucially informs the design considerations related to the electron beam's path.

Gravity’s Effect on Electrons

Gravity affects all objects with mass, even electrons, causing them to accelerate downwards. Although this effect is almost negligible for electrons over short distances, it is still vital to understand how it influences their motion.

• The gravitational force results in a vertical acceleration of \(9.8 \mathrm{~m/s}^2\).
• This acceleration impacts the vertical displacement over time.
• For electrons in a computer monitor setting, the small distance they travel means gravity’s impact is minimal.
• In the given example, the vertical displacement is approximately \(2.66 \times 10^{-14} \text{ m}\), indicating a nearly flat trajectory over short distances.

Thus, designers usually ignore gravitational effects in applications like electron beams within monitors, ensuring the electron’s path remains predominantly horizontal.

Time of Flight Calculation

Understanding the time of flight is key to analyzing the motion of projectiles. For the electron, this is the time it takes to hit the screen once ejected from the electron gun.
Use the horizontal distance and horizontal velocity components to determine the time: \(t = \frac{d}{v_x}\). Here,- \(d\) is the distance to the screen \(0.35 \text{ m}\),- \(v_x\) is the horizontal velocity \(1.5 \times 10^6 \text{ m/s}\).The result \(t \approx 2.33 \times 10^{-7} \text{ s}\) tells us how long the electron is in motion horizontally.

• This calculation assumes constant horizontal velocity and no impediments.
• It's critical for determining how vertical displacement changes during this period.

Ultimately, knowing the time of flight allows designers to fine-tune the system for optimal performance, understanding fully the brief interval in which electrons travel and hit the screen.