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Velocity

CSCA Velocity study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Velocity

1. Core Concepts: Velocity vs. Speed

In Physics (especially for CSCA exams), **Velocity** and **Speed** are distinct concepts. Never use them interchangeably.

| Aspect | Velocity ($\vec{v}$) | Speed ($v$) |

| :--- | :--- | :--- |

| **Definition** | Rate of change of position ($\Delta \vec{x} / \Delta t$) | Rate of distance traveled ($d / \Delta t$) |

| **Nature** | **Vector**: Magnitude + Direction | **Scalar**: Magnitude only |

| **Direction** | Same as **direction of motion** (tangent to path) | None |

| **Meaning** | How fast and where it's going | Just how fast it's moving |

2. Average vs. Instantaneous Velocity

#### A. Average Velocity

* **Definition**: The displacement divided by the time interval.

* **Formula**: $$\vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t} = \frac{\vec{x}_f - \vec{x}_i}{t_f - t_i}$$

* **Graphical Meaning**: On a Position-Time ($x-t$) graph, average velocity is the **slope of the secant line** connecting the start and end points.

#### B. Instantaneous Velocity

* **Definition**: The velocity at a specific instant (limit as $\Delta t \to 0$).

* **Formula**: $$\vec{v} = \lim_{\Delta t \to 0} rac{\Delta \vec{x}}{\Delta t} = \frac{d\vec{x}}{dt}$$

* **Graphical Meaning**: On an $x-t$ graph, instantaneous velocity is the **slope of the tangent line** at that specific point.

3. Critical Trap: Average Speed $\neq$ Magnitude of Average Velocity

This is a common exam trick.

* **Average Speed** = **Total Distance** / Total Time

* **Magnitude of Avg. Velocity** = |**Total Displacement**| / Total Time

**Conclusion**: Unless the motion is **unidirectional and linear**, Average Speed is generally **greater than** the magnitude of Average Velocity. (e.g., A lap around a track has 0 velocity but high speed).

4. Composition of Motion (Boat Crossing River)

Velocity follows the **Parallelogram Rule** for vector addition. This is essential for 2D motion.

* **Model**: Boat velocity relative to water ($\vec{v}_{boat}$) and Water flow velocity ($\vec{v}_{flow}$).

* **Resultant Velocity**: $\vec{v}_{actual} = \vec{v}_{boat} + \vec{v}_{flow}$ (Vector Sum).

* **Independence Principle**: The time to cross the river depends **only** on the velocity component perpendicular to the bank. It is independent of the water flow speed.

$$ t = \frac{\text{Width } d}{v_{boat} \text{ (perpendicular component)}} $$

5. Typical Examples

**Example 1: Graphical Interpretation**

A particle moves along the x-axis. On its $x-t$ graph (parabola), at $t=2s$, the slope of the tangent line is $4$. The slope of the line connecting $t=0$ and $t=2s$ is $2$.

**Find**: (1) Instantaneous velocity at $t=2s$; (2) Average velocity over first 2 seconds.

**Solution**:

(1) Instantaneous velocity = Slope of tangent = $4 \, \text{m/s}$.

(2) Average velocity = Slope of secant = $2 \, \text{m/s}$.

**Example 2: Boat Crossing River**

$v_{boat} = 4 \, \text{m/s}$, $v_{water} = 3 \, \text{m/s}$, River Width = $120 \, \text{m}$.

(1) **Minimum Time**: Point boat perpendicular to bank.

$$ t_{min} = \frac{120}{4} = 30 \, \text{s} $$

(2) **Resultant Velocity**: $\vec{v} = \sqrt{4^2 + 3^2} = 5 \, \text{m/s}$ (directed downstream).