Problem 27 - Fundamentals of Physics

Question

An electron has a constant acceleration of \(+3.2 \mathrm{~m} / \mathrm{s}^{2}\). At a certain instant its velocity is \(+9.6 \mathrm{~m} / \mathrm{s}\). What is its velocity (a) \(2.5 \mathrm{~s}\) earlier and (b) \(2.5 \mathrm{~s}\) later?

Step-by-Step Solution

Answer

(a) 1.6 m/s earlier; (b) 17.6 m/s later.

Step 1: Understanding the Problem

Given an electron with a constant acceleration of \(+3.2 \ \mathrm{m/s^2}\) and a velocity at an instant of \(+9.6 \ \mathrm{m/s}\). We need to find its velocity (a) \(2.5 \ \mathrm{s}\) earlier and (b) \(2.5 \ \mathrm{s}\) later.

Step 2: Using the Formula for Velocity

The formula to calculate the velocity due to constant acceleration is given by:\[ v = u + at \]Where:- \(v\) is the final velocity,- \(u\) is the initial velocity,- \(a\) is the acceleration, and- \(t\) is the time interval.

Step 3: Finding Velocity 2.5 seconds Earlier (Part a)

We need to find the initial velocity \(u\) when the electron's velocity \(v\) is \(+9.6 \ \mathrm{m/s}\), and the time \(t\) is \(-2.5 \ \mathrm{s}\) (since it is earlier, time is negative). Using the formula:\[ 9.6 = u + (3.2)(-2.5) \]Solving for \(u\):\[ u = 9.6 - 8 = 1.6 \ \mathrm{m/s} \]

Step 4: Finding Velocity 2.5 seconds Later (Part b)

Now, we calculate the velocity \(v\) 2.5 seconds later using the same formula with time \(t = +2.5 \ \mathrm{s}\):\[ v = 9.6 + (3.2)(2.5) \]Calculating \(v\):\[ v = 9.6 + 8 = 17.6 \ \mathrm{m/s} \]

Step 5: Solution Compilation

The electron's velocity was \(1.6 \ \mathrm{m/s}\) \(2.5 \ \mathrm{s}\) earlier and will be \(17.6 \ \mathrm{m/s}\) \(2.5 \ \mathrm{s}\) later.

Key Concepts

Acceleration

Acceleration is an essential concept in kinematics and helps us understand how the velocity of an object changes over time. It is defined as the rate at which an object's velocity changes. In the exercise given, the electron experiences a constant acceleration of \(+3.2 \, \mathrm{m/s^2}\).
This means the electron's speed is increasing by \(3.2 \, \mathrm{m/s}\) every second. Constant acceleration implies that the rate of change of velocity remains the same over time. This simplifies the calculation, as we do not need to account for changing acceleration.
To calculate how this acceleration affects velocity, the formula \(v = u + at\) is used. Here, \(v\) is the velocity at a later time, \(u\) is the initial velocity, \(a\) is constant acceleration, and \(t\) is the time interval during which the acceleration acts.

• Positive acceleration increases the velocity.
• Negative acceleration (deceleration) decreases the velocity.

Understanding acceleration helps predict how quickly something like an electron can speed up or slow down over time, making it a crucial aspect of motion analysis.

Velocity

Velocity is a vector quantity that describes the speed of an object in a particular direction. In kinematics, it is crucial to differentiate between speed (a scalar) and velocity, which also considers direction. In the problem, the electron's initial velocity is given as \(+9.6 \, \mathrm{m/s}\), indicating not only how fast the electron is moving but also in which direction.
When discussing changes in velocity, we rely on the formula \(v = u + at\). This equation derives from the basics of motion and helps calculate the velocity at any given time by considering the initial velocity, acceleration, and time.
The steps to find the velocity at different time intervals involve:

• Identifying the initial velocity \(u\).
• Adding the product of acceleration \(a\) and time \(t\) to it.
• Determining changes as they occur before or after the given instant.

This approach helps us quantify the dynamic nature of the electron's path by showing exactly how much the velocity changes over time intervals.

Time Interval

A time interval in kinematics refers to the duration over which motion is observed or measured. In the context of the exercise problem, it involves finding the velocity of an electron at two different points in time - specifically \(2.5 \, \mathrm{s}\) before and \(2.5 \, \mathrm{s}\) after a given instant of velocity measurement.
The concept of time interval is integral as it affects how we calculate changes in motion parameters. It serves as the multiplier for acceleration in the formula \(v = u + at\). Here, the sign of the time interval varies depending on whether we are looking into the past or the future:

• A positive \(t\) implies a calculation into a future velocity at a later time.
• A negative \(t\) implies looking back at a previous velocity, providing a way to "rewind" motion.

This ability to move forward or backward mathematically allows us to predict or revisit motion characteristics accurately. Understanding time intervals in calculations lets us understand how objects like electrons behave under different conditions over time.