chapter 2 key equations (page 81)
write the following equations according to the textbook.
write the equation for displacement and define all the variables (use the variable ( x ) instead of ( d )).
write the equation for average speed and define all the variables.
write the equation for average velocity and define all the variables.
write the equation for acceleration and define all the variables.

Question

Accepted Answer

### 1. Displacement Equation
# Explanation:
## Step1: Recall Displacement Formula
Displacement ($x$) is the change in position. The formula for displacement when an object moves with an average velocity ($v_{avg}$) over a time interval ($t$) is $x = v_{avg} \times t$. If considering initial ($x_0$) and final ($x_f$) positions, $x = x_f - x_0$ (where $x$ here represents displacement, sometimes written as $\Delta x=x_f - x_0$, but following the problem's instruction to use $x$ for displacement instead of $d$, we can present the formula for displacement in terms of average velocity and time as the main formula for linear motion with constant average velocity).
## Step2: Define Variables
- $x$: Displacement (change in position of the object, in units like meters, kilometers, etc.).
- $v_{avg}$: Average velocity (total displacement divided by total time, in units like m/s, km/h, etc.).
- $t$: Time interval (the duration for which the object is moving, in units like seconds, hours, etc.).
Or, if using initial and final position:
- $x_0$: Initial position (position of the object at the start of the time interval, in the same length unit as $x$).
- $x_f$: Final position (position of the object at the end of the time interval, in the same length unit as $x$). And $x=x_f - x_0$ (displacement is the difference between final and initial position).

# Answer:
Equation for Displacement (using average velocity and time): $x = v_{avg} \times t$
Variables:
- $x$: Displacement (change in position of the object).
- $v_{avg}$: Average velocity of the object.
- $t$: Time interval during which the object moves.

(Or, for position - based displacement: $x=x_f - x_0$, where $x_0$ is initial position, $x_f$ is final position, and $x$ is displacement.)

### 2. Average Speed Equation
# Explanation:
## Step1: Recall Average Speed Formula
Average speed ($s_{avg}$) is defined as the total distance traveled ($x_{total}$) divided by the total time taken ($t_{total}$). So the formula is $s_{avg}=\frac{x_{total}}{t_{total}}$.
## Step2: Define Variables
- $s_{avg}$: Average speed (rate of total distance traveled over total time, in units like m/s, km/h, etc.).
- $x_{total}$: Total distance traveled by the object (sum of all path lengths covered, in units like meters, kilometers, etc.).
- $t_{total}$: Total time taken to travel the total distance (sum of all time intervals, in units like seconds, hours, etc.).

# Answer:
Equation for Average Speed: $s_{avg}=\frac{x_{total}}{t_{total}}$
Variables:
- $s_{avg}$: Average speed of the object.
- $x_{total}$: Total distance traveled by the object.
- $t_{total}$: Total time taken to travel the total distance.

### 3. Average Velocity Equation
# Explanation:
## Step1: Recall Average Velocity Formula
Average velocity ($v_{avg}$) is the displacement ($x$) divided by the time interval ($t$) during which the displacement occurs. The formula is $v_{avg}=\frac{x}{t}$, where $x$ can also be written as $x = x_f - x_0$ (displacement as the difference between final and initial position).
## Step2: Define Variables
- $v_{avg}$: Average velocity (vector quantity representing the rate of change of position, in units like m/s, km/h, etc., with direction considered).
- $x$: Displacement (change in position, $x=x_f - x_0$, in units like meters, kilometers, etc.).
- $t$: Time interval (duration for which the displacement occurs, in units like seconds, hours, etc.).
- $x_0$: Initial position (position at the start of the time interval).
- $x_f$: Final position (position at the end of the time interval).

# Answer:
Equation for Average Velocity: $v_{avg}=\frac{x}{t}$ (or $v_{avg}=\frac{x_f - x_0}{t}$)
Variables:
- $v_{avg}$: Average velocity of the object.
- $x$: Displacement of the object ($x = x_f - x_0$).
- $t$: Time interval during which the displacement occurs.
- $x_0$: Initial position of the object.
- $x_f$: Final position of the object.

### 4. Acceleration Equation
# Explanation:
## Step1: Recall Acceleration Formula
Acceleration ($a$) is the rate of change of velocity. The formula for average acceleration is $a=\frac{\Delta v}{t}=\frac{v_f - v_i}{t}$, where $\Delta v$ is the change in velocity, $v_f$ is final velocity, $v_i$ is initial velocity, and $t$ is the time interval.
## Step2: Define Variables
- $a$: Acceleration (rate of change of velocity, in units like m/s², km/h², etc.).
- $v_i$: Initial velocity (velocity of the object at the start of the time interval, in units like m/s, km/h, etc.).
- $v_f$: Final velocity (velocity of the object at the end of the time interval, in units like m/s, km/h, etc.).
- $t$: Time interval (duration over which the velocity changes, in units like seconds, hours, etc.).
- $\Delta v$: Change in velocity ($\Delta v=v_f - v_i$).

# Answer:
Equation for Acceleration: $a=\frac{v_f - v_i}{t}$ (or $a=\frac{\Delta v}{t}$ where $\Delta v = v_f - v_i$)
Variables:
- $a$: Acceleration of the object.
- $v_i$: Initial velocity of the object.
- $v_f$: Final velocity of the object.
- $t$: Time interval during which the velocity changes.
- $\Delta v$: Change in velocity ($\Delta v=v_f - v_i$).

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QUESTION IMAGE

Question

Explanation:

1. Displacement Equation

Step1: Recall Displacement Formula

Step2: Define Variables

Step1: Recall Average Speed Formula

Step2: Define Variables

Step1: Recall Average Velocity Formula

Step2: Define Variables

Answer:

2. Average Speed Equation