91Ó°ÊÓ

The data in the following table represent the initial and final velocities for a boat traveling along the \(x\) axis. The elapsed time for each of the four pairs of velocities in the table is \(2.0 \mathrm{s}\). Review the concept of average acceleration in Section 2.3 and then determine the average acceleration (magnitude and direction) for each of the four pairs. Note that the algebraic sign of your answers will convey the direction. $$\begin{array}{lcc} & \text { Initial velocity } v_{0} & \text { Final velocity } v \\\\\hline \text { (a) } & +2.0 \mathrm{m} /\mathrm{s} & +5.0 \mathrm{m} / \mathrm{s} \\\\\hline \text { (b) } & +5.0 \mathrm{m} / \mathrm{s} & +2.0 \mathrm{m} / \mathrm{s} \\\\\hline \text { (c) } & -6.0 \mathrm{m} / \mathrm{s} & -3.0 \mathrm{m} / \mathrm{s} \\ \hline \text { (d) } & +4.0 \mathrm{m} / \mathrm{s} & -4.0 \mathrm{m} / \mathrm{s} \\\\\hline\end{array}$$

Step by step solution

01

Understand the Concept of Average Acceleration

Average acceleration is defined as the change in velocity divided by the time taken for the change. It is given by the formula \( a = \frac{v - v_0}{t} \), where \( v \) is the final velocity, \( v_0 \) is the initial velocity, and \( t \) is the time taken.

02

Calculate Average Acceleration for Pair (a)

For this pair, the initial velocity \( v_0 = +2.0\, \text{m/s} \) and the final velocity \( v = +5.0\, \text{m/s} \). The time \( t = 2.0\, \text{s} \). Substitute these values into the formula: \[a = \frac{+5.0 - (+2.0)}{2.0} = \frac{3.0}{2.0} = +1.5 \, \text{m/s}^2\]The positive sign indicates the acceleration is in the positive direction.

03

Calculate Average Acceleration for Pair (b)

For this pair, the initial velocity \( v_0 = +5.0\, \text{m/s} \) and the final velocity \( v = +2.0\, \text{m/s} \). Substitute into the formula: \[a = \frac{+2.0 - (+5.0)}{2.0} = \frac{-3.0}{2.0} = -1.5 \, \text{m/s}^2\]The negative sign indicates the acceleration is in the negative direction.

04

Calculate Average Acceleration for Pair (c)

For this pair, the initial velocity \( v_0 = -6.0\, \text{m/s} \) and the final velocity \( v = -3.0\, \text{m/s} \). Substitute into the formula: \[a = \frac{-3.0 - (-6.0)}{2.0} = \frac{3.0}{2.0} = +1.5 \, \text{m/s}^2\]The positive sign indicates the acceleration is in the positive direction.

05

Calculate Average Acceleration for Pair (d)

For this pair, the initial velocity \( v_0 = +4.0\, \text{m/s} \) and the final velocity \( v = -4.0\, \text{m/s} \). Substitute into the formula:\[a = \frac{-4.0 - (+4.0)}{2.0} = \frac{-8.0}{2.0} = -4.0 \, \text{m/s}^2\]The negative sign indicates the acceleration is in the negative direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial Velocity

Initial velocity is the speed at which an object starts its journey. It's the velocity when the time interval considered begins. In the context of this exercise, initial velocity is essential as it acts as a reference point for determining the change in velocity. If a boat's motion is being analyzed along the x-axis, we often denote initial velocity by \( v_0 \). This parameter varies in different problems and might have a positive or negative sign, indicating its direction.

Understanding initial velocity is crucial for calculating acceleration. For instance, if the initial velocity is expressed as \( v_0 = +2.0 \, \text{m/s} \), this means that the boat moves forward at this speed before any changes. Similarly, if \( v_0 = -6.0 \, \text{m/s} \), it suggests that the boat starts moving backward.

**Key Points to Remember:**

• The initial velocity sets the stage for velocity change and determines the sign used in calculations.
• It's typically provided in meters per second (m/s).
• A positive sign indicates motion in the positive x-direction, while a negative sign suggests a reverse motion.

Final Velocity

Final velocity is the speed at which an object ends its journey for the specified time interval. It's what you get after any acceleration or deceleration has been accounted for during motion. In calculations, we usually denote final velocity by \( v \). Like initial velocity, the final velocity can also have positive or negative values indicating direction.

When trying to understand motion and calculate average acceleration, knowing the final velocity is essential. For instance, if a problem states that \( v = +5.0 \, \text{m/s} \), after a time interval, it signifies the boat accelerated forward. On the other hand, if \( v = -4.0 \, \text{m/s} \), it means the boat has moved backward or decelerated to a stop and then reversed.

**Points to Note:**

• Final velocity is impacted by forces acting on the object, showing the result of these influences over the time period.
• It reflects the actual speed and direction of the object at the end of the time interval.
• Calculating the change in velocity involves subtracting the initial velocity from the final velocity.

Acceleration Direction

Direction is a key part of understanding acceleration. Acceleration is a vector quantity, so it has both a magnitude and a direction. The direction of acceleration tells you how the velocity changes over time for an object.

In practice, the acceleration can be in the same direction as the motion (positive) or opposite to the motion (negative). When solving for the average acceleration using the formula \( a = \frac{v - v_0}{t} \), the sign of the result reveals the direction. A positive acceleration (+) indicates an increase in velocity in the forward direction, while a negative acceleration (-) indicates a decrease, or a reversal, in velocity.

**Understanding Acceleration Direction:**

• A positive sign indicates that the object is speeding up in the positive x-direction.
• A negative sign signifies the object is slowing down or moving in the opposite direction.
• Recognizing the direction of acceleration helps predict future motion and understand the forces at play.