91Ó°ÊÓ

When considering projectile motion, it's important to differentiate between horizontal and vertical components. Horizontal motion is defined by a constant velocity, as there is no horizontal acceleration (assuming air resistance is negligible). In our scenario, the electron is projected with a speed of \(3.0 \times 10^{6} \, \text{m/s}\) and needs to cover a horizontal distance of \(1.0\, \text{m}\). To find out how long this takes, we use the formula for constant velocity:

• \( t = \frac{d}{v} \)

where:

• \(d\) is the horizontal distance
• \(v\) is the horizontal velocity

Plugging in the values, we calculate:

• \( t = \frac{1.0}{3.0 \times 10^{6}} \approx 3.33 \times 10^{-7} \, \text{s}\)

This very short time frame highlights the electron's rapid movement across the horizontal plane.

While the electron moves horizontally, it's also affected by vertical forces, particularly due to gravity. Vertical displacement in projectile motion is calculated using the equation:

• \( s = \frac{1}{2} g t^2 \)

where:

• \(s\) is the vertical displacement
• \(g\) is the acceleration due to gravity, typically \(9.8 \, \text{m/s}^2\)
• \(t\) is the time of flight, which we've already calculated as \(3.33 \times 10^{-7} \, \text{s}\)

Substituting these values, we compute:

• \( s = \frac{1}{2} \times 9.8 \times (3.33 \times 10^{-7})^2 \approx 5.44 \times 10^{-13} \, \text{m}\)

This result shows the infinitesimal fall the electron experiences over such a brief span.

A key factor impacting vertical motion is gravity, which consistently acts on objects in projectile motion. In this context, the acceleration due to gravity, denoted as \(g\), is \(9.8 \, \text{m/s}^2\). This force is what causes the electron to fall as it travels horizontally. Importantly, while the horizontal velocity influences the time duration for which gravity acts, gravity itself does not alter horizontal motion.
Increasing the initial horizontal speed decreases the time \(t\), as seen with the inverse relationship in the formula \(t = \frac{d}{v}\). Consequently, with a shorter \(t\), the vertical displacement \(s\) becomes smaller since:

• \( s = \frac{1}{2} g t^2 \)

This understanding clarifies that while gravity is a vertical force, its effect is indirectly felt in how quickly horizontally moving objects like the electron traverse their path.