**Learning Goal:**Suppose an object has an initial velocity \(\vec{v}_i\) at time \(t_i\) and later, at time \(t_f\), has velocity \(\vec{v}_f\). The fact that the velocity changes tells us that the object undergoes an acceleration during the time interval \(\Delta t = t_f - t_i\). From the definition of acceleration,\[\vec{a} = \frac{\vec{v}_f - \vec{v}_i}{t_f - t_i} = \frac{\Delta \vec{v}}{\Delta t},\]we see that the acceleration vector points in the same direction as the vector \(\Delta \vec{v}\). This vector is the change in the velocity \(\Delta \vec{v} = \vec{v}_f - \vec{v}_i\), so to know which way the acceleration vector points, we have to perform the vector subtraction \(\vec{v}_f - \vec{v}_i\). This Tactics Box shows how to use vector subtraction to find the acceleration vector.**Figure Explanation:**The figure illustrates two velocity vectors, \(\vec{v}_i\) and \(\vec{v}_f\). The initial velocity vector, \(\vec{v}_i\), is represented as an arrow pointing to the right with a specific angle. The final velocity vector, \(\vec{v}_f\), is shown as an arrow pointing further to the right at an upward angle. Both vectors originate from different points that are marked with black dots, indicating the positions of the object at the initial and final times. The change in velocity can be visualized as the difference in direction and length between \(\vec{v}_f\) and \(\vec{v}_i\). **TACTICS BOX 4.1: Finding the Acceleration Vector**To find the acceleration between velocity \( \vec{v}_i \) and velocity \( \vec{v}_f \) (Figure 1), follow these steps:1. **Draw the Velocity Vectors**: Position the velocity vectors \( \vec{v}_i \) and \( \vec{v}_f \) with their tails together. - **Diagram 1**: Shows vectors \( \vec{v}_i \) and \( \vec{v}_f \) positioned tail to tail.2. **Draw the Change in Velocity Vector**: Draw a vector from the tip of \( \vec{v}_i \) to the tip of \( \vec{v}_f \). This vector is \( \Delta \vec{v} \), representing \( \vec{v}_f = \vec{v}_i + \Delta \vec{v} \). - **Diagram 2**: Illustrates the vector \( \Delta \vec{v} \) connecting the tips of \( \vec{v}_i \) and \( \vec{v}_f \), forming a triangle.3. **Identify the Acceleration Vector**: Return to the original motion diagram, and draw a vector at the middle dot in the direction of \( \Delta \vec{v} \); label it \( \vec{a} \). This represents the average acceleration at the midpoint between \( \vec{v}_i \) and \( \vec{v}_f \). - **Diagram 3**: Displays the vector \( \vec{a} \) as an orange arrow indicating the direction and sense of change in velocity.---**Part A****Motion Diagram Explanation**:A motion diagram is shown for an object moving along a curved path. The dots, separated by equal intervals, represent the object's position at three subsequent instants. The vectors \( \vec{v}_1 \) and \( \vec{v}_2 \) illustrate the object's average velocity for the first and second intervals.**Instructions**:Draw the vector \( -\vec{v}_1 \) and the acceleration vector \( \vec{a} \), which represents the change in average velocity over the total time interval.- **Diagram**: Visual interface for selecting and drawing vectors, showing \( \vec{v}_1 \

Name: **Learning Goal:**Suppose an object has an initial velocity \(\vec{v}
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Description: **Learning Goal:**Suppose an object has an initial velocity \(\vec{v}_i\) at time \(t_i\) and later, at time \(t_f\), has velocity \(\vec{v}_f\). The fact that the velocity changes tells us that the object undergoes an acceleration during the time i

The dots are separated by equal intervals,are the veolcity vector of the obejct.

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Physics

Suppose an object has an initial velocity vi at time t; and later, at time t, has velocity vf. The fact that the velocity changes tells us that the object undergoes an acceleration during the time interval At = tft₁. From the definition of acceleration, ā Vf - Vi te-ti Δύ Δt 9

Suppose an object has an initial velocity vi at time t; and later, at time t, has velocity vf. The fact that the velocity changes tells us that the object undergoes an acceleration during the time interval At = tft₁. From the definition of acceleration, ā Vf - Vi te-ti Δύ Δt 9

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Displacement, Velocity and Acceleration

In classical mechanics, kinematics deals with the motion of a particle. It deals only with the position, velocity, acceleration, and displacement of a particle. It has no concern about the source of motion.

Linear Displacement

The term "displacement" refers to when something shifts away from its original "location," and "linear" refers to a straight line. As a result, “Linear Displacement” can be described as the movement of an object in a straight line along a single axis, for example, from side to side or up and down. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Linear displacement is usually measured in millimeters or inches and may be positive or negative.

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Question

**Learning Goal:**

Suppose an object has an initial velocity \(\vec{v}_i\) at time \(t_i\) and later, at time \(t_f\), has velocity \(\vec{v}_f\). The fact that the velocity changes tells us that the object undergoes an acceleration during the time interval \(\Delta t = t_f - t_i\). From the definition of acceleration,

\[
\vec{a} = \frac{\vec{v}_f - \vec{v}_i}{t_f - t_i} = \frac{\Delta \vec{v}}{\Delta t},
\]

we see that the acceleration vector points in the same direction as the vector \(\Delta \vec{v}\). This vector is the change in the velocity \(\Delta \vec{v} = \vec{v}_f - \vec{v}_i\), so to know which way the acceleration vector points, we have to perform the vector subtraction \(\vec{v}_f - \vec{v}_i\). This Tactics Box shows how to use vector subtraction to find the acceleration vector.

**Figure Explanation:**

The figure illustrates two velocity vectors, \(\vec{v}_i\) and \(\vec{v}_f\). The initial velocity vector, \(\vec{v}_i\), is represented as an arrow pointing to the right with a specific angle. The final velocity vector, \(\vec{v}_f\), is shown as an arrow pointing further to the right at an upward angle. Both vectors originate from different points that are marked with black dots, indicating the positions of the object at the initial and final times. The change in velocity can be visualized as the difference in direction and length between \(\vec{v}_f\) and \(\vec{v}_i\).

**TACTICS BOX 4.1: Finding the Acceleration Vector**

To find the acceleration between velocity \( \vec{v}_i \) and velocity \( \vec{v}_f \) (Figure 1), follow these steps:

1. **Draw the Velocity Vectors**: Position the velocity vectors \( \vec{v}_i \) and \( \vec{v}_f \) with their tails together.

- **Diagram 1**: Shows vectors \( \vec{v}_i \) and \( \vec{v}_f \) positioned tail to tail.

2. **Draw the Change in Velocity Vector**: Draw a vector from the tip of \( \vec{v}_i \) to the tip of \( \vec{v}_f \). This vector is \( \Delta \vec{v} \), representing \( \vec{v}_f = \vec{v}_i + \Delta \vec{v} \).

- **Diagram 2**: Illustrates the vector \( \Delta \vec{v} \) connecting the tips of \( \vec{v}_i \) and \( \vec{v}_f \), forming a triangle.

3. **Identify the Acceleration Vector**: Return to the original motion diagram, and draw a vector at the middle dot in the direction of \( \Delta \vec{v} \); label it \( \vec{a} \). This represents the average acceleration at the midpoint between \( \vec{v}_i \) and \( \vec{v}_f \).

- **Diagram 3**: Displays the vector \( \vec{a} \) as an orange arrow indicating the direction and sense of change in velocity.

---

**Part A**

**Motion Diagram Explanation**:

A motion diagram is shown for an object moving along a curved path. The dots, separated by equal intervals, represent the object's position at three subsequent instants. The vectors \( \vec{v}_1 \) and \( \vec{v}_2 \) illustrate the object's average velocity for the first and second intervals.

**Instructions**:

Draw the vector \( -\vec{v}_1 \) and the acceleration vector \( \vec{a} \), which represents the change in average velocity over the total time interval.

- **Diagram**: Visual interface for selecting and drawing vectors, showing \( \vec{v}_1 \

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