91Ó°ÊÓ

Understanding constant acceleration is key in analyzing the motion of particles like electrons. Acceleration is defined as the rate of change of velocity over time. When we say an object has constant acceleration, this means that its rate of velocity change remains the same throughout the motion. For our electron, the constant upward acceleration is given as \(5.30 \times 10^{17} \text{ cm/s}^2\). This huge number reflects how rapidly the electron's velocity is increasing in the vertical direction.

In equations of motion, particularly those involving constant acceleration, we often use formulas derived from Newtonian mechanics. These formulas allow us to calculate displacement, velocity, and time. For example:

• \( v = u + at \) - where \(v\) is the final velocity, \(u\) is the initial velocity, and \(a\) is the acceleration
• \( s = ut + \frac{1}{2}at^2 \) - where \(s\) is the displacement, \(u\) is the initial speed, \(t\) is time, and \(a\) is acceleration

Using these equations, you can predict the future position and velocity of an object under constant acceleration. It simplifies calculations significantly, making problems like our electron's motion much easier to solve.

Horizontal displacement refers to the distance an object travels along the horizontal path. In the context of this exercise, it involves an electron moving horizontally at a constant speed. We calculated the time required for an electron to travel a horizontal distance of \(6.20 \text{ cm}\).

The distance formula for an object moving at constant horizontal velocity is \(d = vt\), where \(d\) is the displacement, \(v\) is the velocity, and \(t\) is the time. Since the horizontal speed \(v\) is \(2.10 \times 10^7 \text{ m/s}\), we can simply rearrange the formula to solve for time:

• \( t = \frac{d}{v} \)

This formula is applicable whenever there is no external force, like gravity, acting in the horizontal direction, allowing for a straightforward calculation based on distance and speed alone.

In our specific case, displacement is the entire horizontal distance traveled by the electron, which, given the velocity and distance supplied, gives a clear understanding of the time needed for the electron to move that distance.

Vertical displacement considers how much an object moves in the vertical direction under the influence of forces such as gravity or, in our case, external electric fields. With the given constant vertical acceleration, we can find the electron's vertical displacement using the formula for motion under constant acceleration: \( y = \frac{1}{2} a t^2 \).

The formula considers that:

• \( a \) is the constant acceleration affecting the electron
• \( t \) is the time over which this acceleration acts
• \( y \) represents the resulting vertical height the electron reaches

The initial vertical velocity is zero in this particular problem, since the electron starts with only horizontal motion. By substituting the given values for \( a \) and \( t \), we calculated that the vertical displacement \( y \) is approximately \(2.3 \text{ cm}\). This result illustrates not only the impact of acceleration on motion but also how small changes in time can lead to observable changes in position when operating at high accelerations.