Question

An electron, starting from rest and moving with a constant acceleration, travels 1.0 cm in 2.0 ms. What is the magnitude of this acceleration? a) \(25 \mathrm{~km} / \mathrm{s}^{2}\) b) \(20 \mathrm{~km} / \mathrm{s}^{2}\) c) \(23.0 \mathrm{~m} / \mathrm{s}\) west d) \(23.0 \mathrm{~m} / \mathrm{s}\) east e) \(11.0 \mathrm{~m} / \mathrm{s}\) south

Short answer

Question: Calculate the constant acceleration of an electron starting from rest if it travels 1.0 cm in a time of 2.0 ms. Express the acceleration in km/s². Answer: The constant acceleration of the electron is 5 km/s².

Step-by-step solution

Convert the given quantities into SI units

We need to convert the distance and time into SI units (meters and seconds): Distance: 1.0 cm = 0.01 m Time: 2.0 ms = 0.002 s

Substitute given values into the equation

Now we substitute the given values into the equation for the distance traveled: $$ 0.01 = 0 \cdot 0.002 + \frac{1}{2}a(0.002)^2 $$

Simplify the equation

Simplify the equation to isolate the acceleration variable (a): $$ 0.01 = 0.000002a $$

Solve for the acceleration

Divide both sides of the equation by 0.000002 to get the value of acceleration (a): $$ a = \frac{0.01}{0.000002} = 5000 \mathrm{~m} / \mathrm{s}^{2} $$

Convert acceleration to km/s²

Convert the acceleration from m/s² to km/s²: $$ a = 5000 \mathrm{~m} / \mathrm{s}^{2} = 5 \mathrm{~km} / \mathrm{s}^{2} $$ The magnitude of the electron's acceleration is 5 km/s². None of the answer choices matches this value, so there might be an error in the given options.

Key concepts

Constant Acceleration

When we talk about constant acceleration in physics, we refer to an object that’s increasing its velocity by the same amount in every second of its motion. This is a fundamental concept when studying motion, as it applies to many real-world situations, including the movement of charged particles such as electrons.

Understanding constant acceleration is crucial for predicting how an object will move. For an electron moving in a vacuum or a negligible field, ignoring quantum effects, it can accelerate uniformly under the influence of an electric field. This is because the electric field exerts a steady force, leading to a constant rate of acceleration, provided no other forces act upon it.

The idea that the electron in the problem is moving with constant acceleration means that we can apply certain kinematic equations formulated for such conditions directly. These equations help us calculate various parameters of motion like final velocity, time taken, displacement, and in this case, acceleration, making constant acceleration a pivotal concept in kinematics.

SI Units Conversion

Converting to and from SI units, which are the international system of units, is a fundamental skill in physics and many other sciences. The importance of converting units to SI (Standard International) before solving physics problems cannot be overstated, as it ensures that equations and formulas work as intended since they are typically derived using these units.

For instance, in the provided exercise, distances are measured in meters (m) and time in seconds (s). Seeing that the original measurements were given in centimeters (cm) for distance and milliseconds (ms) for time, they are not in SI units and therefore need to be converted. This process involves moving decimal points or multiplying by conversion factors:

• To convert from centimeters to meters, we divide by 100 (since there are 100 centimeters in a meter).
• To convert from milliseconds to seconds, we divide by 1,000 (as there are 1,000 milliseconds in a second).

Performing these conversions correctly is crucial for obtaining the correct results in physics problems.

Kinematic Equations

Kinematic equations, also known as the equations of motion, are essential tools in physics for linking the various parameters of an object's motion, specifically when the motion is uniformly accelerated. These equations relate displacement, initial velocity, final velocity, acceleration, and time.

The kinematic equation used in the step-by-step solution looks like this: \[ d = v_i t + \frac{1}{2}at^2 \
It represents the distance (\(d\)) an object travels when starting with an initial velocity (\(v_i\right)) over a certain time (\(t\)), and with constant acceleration (\(a\right)). Since the electron starts from rest, its initial velocity is zero, and the equation simplifies to: \[ d = \frac{1}{2}at^2 \
These kinematic equations are derived from the definitions of velocity and acceleration. Moreover, they can only be used when acceleration is constant throughout the duration of the motion, which is an assumed condition of our electron problem.

Understanding and correctly applying kinematic equations not only helps to solve textbook problems but also strengthens the foundation for more complex physics concepts. Keeping track of the initial conditions and the values after certain periods is key to successfully employing these equations for analyzing motion.