An X-ray tube gives electrons constant acceleration over a distance of \(15 \mathrm{cm} .\) If their final speed is \(1.2 \times 10^{7} \mathrm{m} / \mathrm{s},\) what are (a) the electrons' acceleration and (b) the time they spend accelerating?

Understanding constant acceleration is foundational to solving various kinematics problems. Constant acceleration occurs when an object speeds up or slows down at the same rate throughout a period of motion. This means that the acceleration remains unchanged for the duration that the object is in motion.

When dealing with problems involving constant acceleration, such as the acceleration of an electron in an X-ray tube, it’s useful to apply specific kinematic equations formulated for constant acceleration scenarios. In this textbook exercise, the electron's speed changes from zero to a final velocity of over a distance, all under constant acceleration.

One of the typical features of motion under constant acceleration is that all the motion parameters—displacement, velocity, and time—can be interrelated using kinematic equations. The calculation to find the electron's acceleration, given the final speed and the distance over which this speed was reached, is a clear application of the concepts of constant acceleration.

Kinematic equations are the tools that physicists use to link velocity, acceleration, time, and displacement. These equations are only applicable under the assumption of constant acceleration and provide a robust framework for solving various physics problems.

The first kinematic equation, often indicated as \( v^2 = u^2 + 2as \), helps to find an object's final velocity based on its initial velocity, acceleration, and displacement. In the electron acceleration problem, we use this equation to find the acceleration, given the final velocity and the distance. The equation is rearranged to \( a = \frac{v^2}{2s} \).

Another kinematic equation, \( v = u + at \), helps to determine the time it takes for the object to reach the final velocity, which is extremely helpful in the context of the provided exercise. By setting the initial velocity to zero and rearranging the equation, we can calculate the total time taken for the electron to achieve its final speed by using \( t = \frac{v}{a} \).

In contexts such as X-ray tubes, electron acceleration is a critical concept. Electrons in these tubes are exposed to electric fields, which provide the constant acceleration needed for them to achieve high speeds. It’s an example of how electric fields can do work on charged particles, increasing their kinetic energy and thus their speed.

In the exercise, the electron is accelerated from rest to a remarkably high speed in a short distance, representing an incredibly rapid acceleration. The significant part of this problem is understanding that the acceleration equation is derived from fundamental principles and how the electron’s final speed can yield insights into the characteristics of the acceleration it experienced.

Overall, the concepts of constant acceleration and kinematic equations are deeply intertwined in problems like electron acceleration. These principles are not limited to massive objects but also apply to subatomic particles, demonstrating the breadth of utility these equations hold in physics.