A.1.1 Describing motion quantitatively and qualitatively

Scalar and Vector Quantities

Scalar: Magnitude Only

It tells you how much of something there is, but not where or in which direction.

Common examples include:

1. Distance: The total length of the path traveled, regardless of direction.
2. Speed: How fast an object is moving, without considering its direction.
3. Time: A measure of duration.

Vectors: Magnitude and Direction

This makes it more descriptive than a scalar. Examples include:

1. Displacement: The straight-line distance between an object’s starting and ending points, along with the direction.
2. Velocity: The rate of change of displacement, including direction.
3. Acceleration: The rate of change of velocity, including direction.

The Role of Projections

1. While vector quantities contain both magnitude and direction, many physical analyses require expressing how much of a vector acts along a specific axis, such as the x-axis or y-axis.
2. This is done through a process called projection, which transforms a vector into a scalar component aligned with a chosen direction.

Calculating and Using Scalar Projections

To determine the scalar projection of a vector onto a chosen axis, one must identify the angle between the vector and the axis and apply the cosine or sine function.

1. In Cartesian coordinates, if the axis of interest aligns with, for example, the $x$-axis, then the scalar projection corresponds simply to the $x$-component of the vector.
   1. This is typically expressed as $A_x=|\overrightarrow{A}|\cos \theta$, where $\theta$ is the angle between $\overrightarrow{A}$ and the positive $x$-direction.
2. If the axis of interest is the $y$-axis, then the scalar projection corresponds to the $y$-component of the vector.
   1. Correspondingly, This is typically expressed as $A_y=|\overrightarrow{A}|\sin \theta$, where $\theta$ is the angle between the vector $\overrightarrow{A}$ and the positive $x$-axis.

A scalar projection represents the magnitude of the vector's influence along an axis and carries a sign that indicates direction.

1. A positive projection means the vector points in the same direction as the defined positive axis.
2. A negative projection indicates it points in the opposite direction.

Comparing Scalars and Vectors

Speed and Velocity: Comparing Two Measures of Motion

Average Speed and Average Velocity

Both speed and velocity describe how fast something is moving, but they differ in their treatment of direction.

It is expressed by:

$$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$$

Or symbolically:

$$\bar{v}=\frac{s_{total}}{t_{total}}$$

It is expressed by:

$$\text{Average Velocity}=\frac{\text{Total Displacement}}{\text{Total Time}}$$

Once again, symbolically:

$${\overrightarrow{v}}_{avg}=\frac{\mathrm{\Delta }\overrightarrow{s}}{\mathrm{\Delta }t}$$

Instantaneous Speed and Instantaneous Velocity

While average speed and velocity describe motion over a period of time, instantaneous speed and instantaneous velocity describe motion at a specific moment.

1. To find instantaneous speed/velocity, we thus want to evaluate the speed/velocity at a particular instant of time (over very short time period).
2. Mathematically that corresponds to taking a limit of the average speed/velocity as the time interval becomes infinitely small $\mathrm{\Delta }t\to 0$: $$v(t)=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{\mathrm{\Delta }s}{\mathrm{\Delta }t}=\frac{ds}{dt}$$ $$\overrightarrow{v}(t)=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{\mathrm{\Delta }\overrightarrow{s}}{\mathrm{\Delta }t}=\frac{d\overrightarrow{s}}{dt}$$
3. Which essentially, by definition, is the same as taking the derivative with respect to time.
4. Thus, if you are given a graph of a function that represents the coordinate/position of an object with respect to time, speed at some moment in time can be found by calculating slope of tangent at that particular time moment.

Acceleration: The Rate of Change of Velocity

Acceleration measures how quickly an object’s velocity changes.

Average Acceleration

From definition, it follows, that the average acceleration is:

$$\text{Average Acceleration}={\overrightarrow{a}}_{avg}=\frac{\mathrm{\Delta }\overrightarrow{v}}{\mathrm{\Delta }t}$$

where:

• $\mathrm{\Delta }\overrightarrow{v}$ is the change in velocity (final velocity minus initial velocity).
• $\mathrm{\Delta }t$ is the time interval over which the change occurs.

Instantaneous Acceleration

It’s determined by the slope of the velocity-time graph at that point.

Higher Level content

Similarly as with instantaneous velocity, we consider the average acceleration over very short time period $\mathrm{\Delta }t\to 0$:

$$a(t)=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}=\frac{dv}{dt}$$

Or, in vector form:

$$\overrightarrow{a}(t)=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{\mathrm{\Delta }\overrightarrow{v}}{\mathrm{\Delta }t}=\frac{d\overrightarrow{v}}{dt}$$

Graphical Representation of Motion

Graphs are powerful tools for visualizing motion. Let’s explore three key types of motion graphs:

Displacement-Time Graphs

1. The slope of a displacement-time graph represents velocity.
2. A line parallel to the time axis indicates zero velocity.
3. A straight line indicates constant velocity.
4. A curved line indicates changing velocity (non-zero acceleration).

Velocity-Time Graphs

1. The slope of a velocity-time graph represents acceleration.
2. The area under the graph represents displacement.
3. A horizontal line indicates constant velocity.
4. A straight line with a positive slope indicates uniformly increasing velocity.
5. A straight line with a negative slope indicates uniformly decelerating velocity.

Acceleration-Time Graphs

1. A horizontal line indicates constant acceleration.
2. A straight line with a positive slope indicated uniformly increasing acceleration.
3. The area under the graph represents the change in velocity.

Appendix

1. Imagine an object moving with constant velocity $v$ for a time $t$.
2. If we draw this on a velocity-time graph, we get a horizontal line at height $v$.
3. The graph forms a rectangle with width $t$ and height $v$. $$\text{Area}=\text{height}\times \text{width}=v\times t=\text{displacement}$$
4. Displacement is the total change in position of an object over a time interval.
5. If the velocity $v(t)$ varies with time, we can approximate the displacement by dividing the interval from $t=t_1$ to $t=t_2$ into $n$ small sub-intervals of equal width $\mathrm{\Delta }t=\frac{t_2-t_1}{n}$.
6. At each small interval, we assume the velocity is roughly constant and calculate a small displacement:

$$\mathrm{\Delta }{s}_i\approx v(t_i^{\ast })\cdot \mathrm{\Delta }t$$
where $t_i^{\ast }$ is a sample point in the $i$-th subinterval.

Adding up all these small displacements gives an approximate total:

$$\mathrm{\Delta }s\approx \sum _{i=1}^{n}v(t_i^{\ast })\cdot \mathrm{\Delta }t$$

To get the exact displacement, we take the limit as $\mathrm{\Delta }t\to 0$ (or equivalently, as $n\to \mathrm{\infty }$):

$$\mathrm{\Delta }s=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}v(t_i^{\ast })\cdot \mathrm{\Delta }t$$

This limit is defined in calculus as the definite integral: $$\mathrm{\Delta }s={\int }_{t_1}^{t_2}v(t)dt$$

This expression gives the exact displacement, even when velocity is changing continuously over time.