Problem 2 - Calculus

Question

The table gives the position of a particle moving along the \(x\) -axis as a function of time in seconds, where \(x\) is in meters. What is the average velocity of the particle from \(t=0\) to \(t=4 ?\) $$\begin{array}{c|c|c|c|c|c}\hline t & 0 & 2 & 4 & 6 & 8 \\\\\hline x(t) & -2 & 4 & -6 & -18 & -14 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The average velocity of the particle is \(-1\, \text{m/s}\).

Step 1: Identify the Time Interval

First, we need to identify the interval over which we're calculating the average velocity. From the problem, we are interested in the interval from \( t = 0 \; \text{seconds} \) to \( t = 4 \; \text{seconds} \).

Step 2: Find Initial and Final Positions

We need the positions at the start and end of the interval. From the table, at \( t = 0 \), the position \( x(0) = -2 \; \text{m} \). At \( t = 4 \), the position \( x(4) = -6 \; \text{m} \).

Step 3: Use the Average Velocity Formula

The average velocity \( v_{\text{avg}} \) is given by the formula: \[ v_{\text{avg}} = \frac{x(t_2) - x(t_1)}{t_2 - t_1} \]Substitute \( t_1 = 0 \) and \( t_2 = 4 \).

Step 4: Calculate Displacement

Calculate the displacement: \[ x(4) - x(0) = -6 - (-2) = -6 + 2 = -4 \; \text{meters} \].

Step 5: Calculate Time Interval

Calculate the time interval:\[ t_2 - t_1 = 4 - 0 = 4 \; \text{seconds} \].

Step 6: Compute Average Velocity

Plug the displacement and the time interval into the average velocity formula:\[ v_{\text{avg}} = \frac{-4 \; \text{m}}{4 \; \text{s}} = -1 \; \text{m/s} \].

Key Concepts

Displacement

Displacement is a key concept when discussing average velocity. It refers to the change in position of an object. Displacement is not concerned with the path taken but only the initial and final positions.

In our exercise, the particle's position changes from \( x(0) = -2 \; \text{m} \) at time \( t = 0 \) to \( x(4) = -6 \; \text{m} \) at time \( t = 4 \).

We calculate the displacement using the formula:\[ \Delta x = x(t_2) - x(t_1) \]For the given problem, the displacement is \(-6 - (-2) = -4 \; \text{meters}\).

This calculation tells us how far the particle moved along the \( x \)-axis, from its initial to its final position.

Time Interval

The time interval is the duration between two points in time where we observe the movement. It is essential for calculating the average velocity because it represents the "when" of displacement.

In the exercise, the time starts at \( t = 0 \; \text{seconds} \) and ends at \( t = 4 \; \text{seconds} \).

To calculate the time interval, use the difference between the final and initial times:\[ \Delta t = t_2 - t_1 \]Here, \( t_2 = 4 \) and \( t_1 = 0 \), yielding a time interval of \( 4 \; \text{seconds} \).

This value is crucial for determining how long the particle took to achieve the displacement, directly influencing the average velocity.

Particle Motion

Particle motion refers to the movement of a particle along a path. In this problem, the particle moves along the \( x \)-axis. We're interested in how this motion translates into different positions over time.

The table provided in the exercise demonstrates the particle's movement at specified times. This movement can be tracked by observing the particle's positions at different time intervals.

• At \( t = 0 \; \text{seconds} \), the position was \( -2 \; \text{m} \).
• At \( t = 4 \; \text{seconds} \), its position was \( -6 \; \text{m} \).

Understanding particle motion is fundamental for solving physics problems about speed and velocity, as it dictates how a particle's position changes in response to time.

Position Function

The position function defines the location of a particle as a function of time. In mathematical terms, it's expressed as \( x(t) \), where \( x \) indicates position along the axis and \( t \) is time.

From the exercise, the position function is implied via a table of values instead of a continuous equation. This means we must rely on discrete data points at given times:

• At \( t = 0 \; \text{seconds} \), \( x(t) = -2 \; \text{m} \).
• At \( t = 2 \; \text{seconds} \), \( x(t) = 4 \; \text{m} \).
• At \( t = 4 \; \text{seconds} \), \( x(t) = -6 \; \text{m} \).

To discover more about the particle's movement, we analyze the change from one position to another over time, as observed in these table values. The position function helps to graph and understand the pattern of movement over time.