Scalar and Vector Quantities
Scalar: Magnitude Only
Scalar quantities
Scalar quantities have only magnitude and no direction.
It tells you how much of something there is, but not where or in which direction.
Common examples include:
- Distance: The total length of the path traveled, regardless of direction.
- Speed: How fast an object is moving, without considering its direction.
- Time: A measure of duration.
For instance, if you walk 5 km in a circle and return to your starting point, your total distance is 5 km, even though you haven’t changed your position.
Vectors: Magnitude and Direction
Vector quantities
Vector quantities have both magnitude and direction.
This makes it more descriptive than a scalar. Examples include:
- Displacement: The straight-line distance between an object’s starting and ending points, along with the direction.
- Velocity: The rate of change of displacement, including direction.
- Acceleration: The rate of change of velocity, including direction.
Returning to the road trip example, if you drive 100 km north and then 100 km east, your displacement is the straight-line distance from your starting point to your final position, along with the direction (northeast).
The Role of Projections
- 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.
- 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.
- 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.
- This is typically expressed as $A_x = |\vec{A}|\cos\theta$, where $\theta$ is the angle between $\vec{A}$ and the positive $x$-direction.
- If the axis of interest is the $y$-axis, then the scalar projection corresponds to the $y$-component of the vector.
- Correspondingly, This is typically expressed as $A_y = |\vec{A}|\sin\theta$, where $\theta$ is the angle between the vector $\vec{A}$ 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.
- A positive projection means the vector points in the same direction as the defined positive axis.
- A negative projection indicates it points in the opposite direction.
- This sign is physically meaningful.
- For instance, a negative projection of acceleration along an axis implies that the acceleration is directed against the chosen positive direction.
- It is an important consideration when analyzing situations such as deceleration or motion reversal.
Understanding projections is essential for correctly applying kinematic and dynamic equations, as it allows us to bridge the directional nature of vectors with the scalar quantities used in one-dimensional calculations.
Comparing Scalars and Vectors
| Quantity | Scalar | Vector |
|---|---|---|
| Distance vs. Displacement | Total path length | Straight-line change in position |
| Speed vs. Velocity | Magnitude of velocity | Speed with direction |
| Energy | Always scalar | N/A |
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.
Average speed
Average speed is the total distance traveled divided by the total time taken. It’s a scalar quantity.
It is expressed by:
$$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $$
Or symbolically:
$$\overline{v} = \frac{s_{total}}{t_{total}}$$
Average velocity
Average velocity is the total displacement divided by the total time taken. It’s a vector quantity.
It is expressed by:
$$ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} $$
Once again, symbolically:
$$\vec{v}_{avg} = \frac{\Delta \vec{s}}{\Delta t}$$
ExampleA car travels 100 km north and then 50 km south in 3 hours.
- Average speed: $$\frac{150 \, \mathrm{km}}{3 \, \mathrm{hours}} = 50 \, \mathrm{km \, h}^{-1}$$
- Average velocity: $$\frac{50 \, \mathrm{km (north)}}{3 \, \mathrm{hours}} = 16.7 \, \mathrm{km \, h}^{-1}$$
A car travels for 2 hours with an average speed of $50 \,\mathrm{km/h}$, and then for 3 hours with an average speed of $70\,\mathrm{km/h}$. Assuming the speeds represent averages over each respective time interval, what's the average speed of the car over the entire trip?
Solution
Since average speed is defined as total distance divided by total time over an interval, we can compute the distance for each part as:
\[d_1 = v_1 \cdot t_1 = 50 \cdot 2 = 100~\mathrm{km}\]
\[d_2 = v_2 \cdot t_2 = 70 \cdot 3 = 210~\mathrm{km}\]
The total distance and total time of the trip are:
\[d_{\text{total}} = d_1 + d_2 = 100 + 210 = 310~\mathrm{km}\]
\[t_{\text{total}} = t_1 + t_2 = 2 + 3 = 5~\mathrm{h}\]
Recall, that the average speed over the entire journey is defined as:
\[v_{\text{avg}} = \frac{d_{\text{total}}}{t_{\text{total}}} = \frac{310}{5} = 62~\mathrm{km/h}\]
- Students typically confuse average speed/velocity and mean speed/velocity.
- Recall that average means over whole distance and time being considered.
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.
ExampleThe speedometer in a car shows your instantaneous speed.
NoteFurther, we present a slightly more advanced treatment than IB syllabus requires, although we believe it deepens the understanding. The prerequisite knowledge assumed: Topic 5 - Calculus.
- To find instantaneous speed/velocity, we thus want to evaluate the speed/velocity at a particular instant of time (over very short time period).
- Mathematically that corresponds to taking a limit of the average speed/velocity as the time interval becomes infinitely small $\Delta t \to 0$: $$ v(t) = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt} $$ $$ \vec{v}(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} = \frac{d\vec{s}}{dt} $$
- Which essentially, by definition, is the same as taking the derivative with respect to time.
