after watching the video, define the following terms and answer some questions: 1. distance 2. displacement 3. instantaneous velocity 4. average velocity 5. in a position - time graph what does a horizontal straight line indicate? 6. in a position - time graph what does a straight line going up represent? 7. in a position - time graph what does a straight line going down represent? 8. in a position - time graph what does a curved line going up indicate? 9. in a position - time graph what does a line that curves down represent? 10. what is the unit of velocity or speed in a position - time graph?

Question

Accepted Answer

### 1. Distance
# Brief Explanations:
Distance is the total length of the path traveled by an object, regardless of direction. It is a scalar quantity, meaning it has only magnitude and no direction. For example, if a person walks 5 meters east and then 3 meters west, the distance traveled is $5 + 3 = 8$ meters.
# Answer:
Distance is the total length of the path an object travels, a scalar quantity with only magnitude.

### 2. Displacement
# Brief Explanations:
Displacement is the straight - line distance between the initial and final positions of an object, along with the direction from the initial to the final position. It is a vector quantity (has both magnitude and direction). Using the previous example, if a person walks 5 meters east and then 3 meters west, the displacement is $5 - 3=2$ meters east.
# Answer:
Displacement is the straight - line distance (with direction) between an object's initial and final positions, a vector quantity.

### 3. Instantaneous velocity
# Brief Explanations:
Instantaneous velocity is the velocity of an object at a specific instant in time. It can be thought of as the limit of the average velocity as the time interval over which the average velocity is calculated approaches zero. Mathematically, if $x(t)$ is the position function of an object with respect to time $t$, the instantaneous velocity $v(t)$ is given by $v(t)=\lim_{\Delta t
ightarrow0}\frac{\Delta x}{\Delta t}=\frac{dx(t)}{dt}$ (the derivative of the position function with respect to time). For example, the speedometer in a car shows the instantaneous speed (the magnitude of instantaneous velocity) at any given moment.
# Answer:
Instantaneous velocity is the velocity of an object at a specific instant, given by $\lim_{\Delta t
ightarrow0}\frac{\Delta x}{\Delta t}$ (or the derivative of position with respect to time).

### 4. Average velocity
# Brief Explanations:
Average velocity is defined as the displacement of an object divided by the time interval during which the displacement occurred. The formula for average velocity $\bar{v}=\frac{\Delta x}{\Delta t}$, where $\Delta x$ is the displacement and $\Delta t$ is the time interval. For instance, if an object has a displacement of 10 meters in 5 seconds, the average velocity is $\frac{10}{5} = 2$ meters per second.
# Answer:
Average velocity is $\bar{v}=\frac{\text{Displacement }(\Delta x)}{\text{Time interval }(\Delta t)}$, the rate of change of displacement.

### 5. In a position - time graph what does a horizontal straight line indicate?
# Brief Explanations:
In a position - time graph, the vertical axis represents position and the horizontal axis represents time. A horizontal straight line means that the position of the object is not changing as time passes. Since velocity is the rate of change of position with respect to time ($v = \frac{\Delta x}{\Delta t}$), if $\Delta x = 0$ (position doesn't change) for any $\Delta t$, the velocity $v=0$. So, a horizontal line in a position - time graph indicates that the object is at rest (velocity is zero).
# Answer:
A horizontal straight line in a position - time graph indicates the object is at rest (velocity $= 0$).

### 6. In a position - time graph what does a straight line going up represent?
# Brief Explanations:
In a position - time graph, the slope of the line is given by $\text{slope}=\frac{\Delta x}{\Delta t}$, which is equal to the velocity of the object. A straight line going up has a positive slope. Since slope $=\frac{\Delta x}{\Delta t}=v$, a positive slope means a positive velocity. So, a straight line going up in a position - time graph represents an object moving with a constant positive velocity (the object is moving away from the origin or in the positive direction with a constant speed).
# Answer:
A straight line going up in a position - time graph represents an object moving with a constant positive velocity.

### 7. In a position - time graph what does a straight line going down represent?
# Brief Explanations:
The slope of a line in a position - time graph is $\frac{\Delta x}{\Delta t}$ (velocity). A straight line going down has a negative slope. Since $v = \frac{\Delta x}{\Delta t}$, a negative slope implies a negative velocity. This means the object is moving with a constant negative velocity (moving towards the origin or in the negative direction with a constant speed).
# Answer:
A straight line going down in a position - time graph represents an object moving with a constant negative velocity.

### 8. In a position - time graph what does a curved line going up indicate?
# Brief Explanations:
In a position - time graph, the slope of the tangent to the curve at a point gives the instantaneous velocity at that point. A curved line going up means that as time increases, the position of the object is increasing. Moreover, if the curve is getting steeper (the slope of the tangent is increasing), the instantaneous velocity is increasing (positive acceleration). If the curve is getting less steep (the slope of the tangent is decreasing), the instantaneous velocity is still positive but decreasing (negative acceleration, but still moving forward). In general, a curved line going up indicates that the object is moving in the positive direction and its velocity is changing (either increasing or decreasing, but the position is still increasing with time).
# Answer:
A curved line going up in a position - time graph indicates the object moves in the positive direction with changing velocity (acceleration is non - zero).

### 9. In a position - time graph what does a line that curves down represent?
# Brief Explanations:
For a position - time graph, the slope of the tangent to the curve is the instantaneous velocity. A line that curves down means that as time increases, the position of the object first increases (since the curve is going up initially) and then may start to increase at a slower rate or even decrease (depending on the curvature). The slope of the tangent (instantaneous velocity) is decreasing. If the slope is positive but decreasing, the object is moving in the positive direction but slowing down. If the slope becomes negative, the object starts moving in the negative direction. Overall, a curved - down line indicates that the object's velocity (the slope of the tangent) is changing, and the rate of increase of position is decreasing (and may eventually become a decrease in position).
# Answer:
A line that curves down in a position - time graph indicates the object's velocity (tangent slope) changes, with the rate of position increase decreasing (may reverse direction).

### 10. What is the unit of velocity or speed in a position - time graph?
# Brief Explanations:
Velocity is defined as the rate of change of position with respect to time ($v=\frac{\Delta x}{\Delta t}$). The SI unit of position ($x$) is the meter ($m$) and the SI unit of time ($t$) is the second ($s$). So, the unit of velocity (or speed, since speed is the magnitude of velocity) is meters per second ($m/s$). Other common units include kilometers per hour ($km/h$) or miles per hour ($mph$), but the SI unit is $m/s$.
# Answer:
The unit of velocity or speed in a position - time graph is meters per second ($m/s$) (or other units like $km/h$, $mph$, etc., but $m/s$ is the SI unit).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1. Distance

Answer:

2. Displacement