91Ó°ÊÓ

The initial velocity \(v_{0}\) and acceleration \(a\) of four moving objects at a given instant in time are given in the table: $$ \begin{array}{|c|c|c|} \hline & \text { Initial velocity } v_{0} & \text { Acceleration } a \\ \hline \text { (a) } & +12 \mathrm{~m} / \mathrm{s} & +3.0 \mathrm{~m} / \mathrm{s}^{2} \\ \hline \text { (b) } & +12 \mathrm{~m} / \mathrm{s} & -3.0 \mathrm{~m} / \mathrm{s}^{2} \\ \hline \text { (c) } & -12 \mathrm{~m} / \mathrm{s} & +3.0 \mathrm{~m} / \mathrm{s}^{2} \\ \hline \text { (d) } & -12 \mathrm{~m} / \mathrm{s} & -3.0 \mathrm{~m} / \mathrm{s}^{2} \\ \hline \end{array} $$ Draw vectors for \(v_{0}\) and \(a\) and, in each case, state whether the speed of the object is increasing or decreasing in time. Account for your answers. Problem For each of the four pairs in the table above, determine the final speed of the object if the elapsed time is \(2.0 \mathrm{~s}\). Compare your final speeds with the initial speeds and make sure that your answers are consistent with your answers to the Concept Questions.

Step by step solution

01

Understanding Initial Conditions

First, observe the initial conditions given for each object. The initial velocity \(v_0\) and acceleration \(a\) for each object is provided. This information will help determine whether the speed is increasing or decreasing. Remember, speed increases if the acceleration is in the same direction as the velocity, and decreases if they oppose each other.

02

Applying Kinematics Equation

Use the kinematics equation for final velocity: \(v = v_0 + a \cdot t\), where \(v\) is the final velocity, \(v_0\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time elapsed. Here, \(t = 2.0\,\text{s}\). Calculate the final velocity for each object by plugging in the given values.

03

Calculate Final Velocity for Object (a)

For object (a), \(v_0 = +12\,\text{m/s}\) and \(a = +3.0\,\text{m/s}^2\). Substituting into the formula gives \(v = 12 + 3 \times 2 = 18\,\text{m/s}\). Since \(v_0\) and \(a\) have the same direction, the speed is increasing.

04

Calculate Final Velocity for Object (b)

For object (b), \(v_0 = +12\,\text{m/s}\) and \(a = -3.0\,\text{m/s}^2\). Substituting into the formula gives \(v = 12 - 3 \times 2 = 6\,\text{m/s}\). Since \(v_0\) and \(a\) are in opposite directions, the speed is decreasing.

05

Calculate Final Velocity for Object (c)

For object (c), \(v_0 = -12\,\text{m/s}\) and \(a = +3.0\,\text{m/s}^2\). Substituting into the formula gives \(v = -12 + 3 \times 2 = -6\,\text{m/s}\). Since \(v_0\) and \(a\) are in opposite directions, the speed is decreasing.

06

Calculate Final Velocity for Object (d)

For object (d), \(v_0 = -12\,\text{m/s}\) and \(a = -3.0\,\text{m/s}^2\). Substituting into the formula gives \(v = -12 - 3 \times 2 = -18\,\text{m/s}\). Since \(v_0\) and \(a\) have the same direction, the speed is increasing.

07

Compare Final and Initial Speeds

Finally, compare the calculated final speeds with the initial speeds: (a) 18 m/s > 12 m/s, (b) 6 m/s < 12 m/s, (c) 6 m/s < 12 m/s, (d) 18 m/s > 12 m/s. This confirms that cases (a) and (d) involve increasing speed, while (b) and (c) involve decreasing speed.

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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 a fundamental concept in kinematics that describes the speed of an object at the starting point of observation. It is represented by the symbol \(v_0\) and indicates how fast and in which direction an object begins its motion. Understanding initial velocity is crucial in predicting how the object will move in the future.
In the exercise, initial velocities vary, with some objects having positive velocities \((+12\,\text{m/s})\) while others have negative \((-12\,\text{m/s})\). A positive initial velocity indicates that the object is moving in the positive direction, while a negative value means it is moving in the opposite direction.
Knowing the initial velocity allows you to apply other kinematics principles, such as acceleration and time, to predict the object's future velocity and movement direction.

Acceleration

Acceleration represents the rate at which an object's velocity changes over time, and it is often denoted by the letter \(a\). This can mean either an increase or decrease in speed or a change in direction. It is crucial to note that acceleration is a vector, meaning it has both a magnitude and a direction.
In this problem, acceleration varies for each object. Some are accelerating at \(+3.0\,\text{m/s}^2\), and others at \(-3.0\,\text{m/s}^2\). A positive acceleration means the object is speeding up in the positive direction, while a negative acceleration means it is either slowing down in the positive direction or speeding up in the negative direction.
Acceleration is an essential factor in determining whether an object's speed is increasing or decreasing over time. By aligning the direction of acceleration and velocity, one can predict the future behavior of the object's motion.

Final Speed

Final speed is the speed an object reaches after it has been moving for a certain period, influenced by its initial velocity and the acceleration it experiences. It's always calculated at a particular point in time after motion has commenced.
In our example, the final speed of each object after \(t = 2.0\,\text{s}\) is determined using the kinematics equation \(v = v_0 + a \cdot t\). For example, for object (a), starting with an initial velocity of \(+12\,\text{m/s}\) and accelerating at \(+3.0\,\text{m/s}^2\), the final speed after 2 seconds is \(18\,\text{m/s}\).
Comparing final speed to initial speed helps determine if an object has sped up or slowed down. This comparison aligns with the observed directions of velocity and acceleration.

Kinematics Equation

The kinematics equation \(v = v_0 + a \cdot t\) is a vital tool in solving motion problems where the initial velocity, acceleration, and time are known. This equation gives a clear way to calculate the final velocity of an object at any given time \(t\).
In our case, by inputting the values for initial velocity \(v_0\), acceleration \(a\), and the elapsed time \(t\), we can solve for the final velocity \(v\). For instance, the motion of object (d) with \(v_0 = -12\,\text{m/s}\) and \(a = -3.0\,\text{m/s}^2\) over 2 seconds results in a final velocity of \(-18\,\text{m/s}\).
Using this equation not only facilitates the calculation of final velocities but also helps in judging whether an object is accelerating in the same direction or in the opposite direction to its motion.