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Acceleration

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

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Acceleration

1. Core Definition

Acceleration is not just about "going faster"; it measures **how fast the velocity changes** and **in what direction**.

* **Definition**: The rate of change of velocity.

* **Formula**: $$\vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{\Delta t}$$

* **Vector Nature**: The direction of $\vec{a}$ is determined solely by the direction of **change in velocity ($\Delta \vec{v}$)**, not necessarily the direction of velocity $\vec{v}$ itself.

2. Core Difficulty: Speeding Up or Slowing Down?

This is the most frequent testing point in CSCA. **Remember: $a < 0$ does NOT mean slowing down!**

The only criterion for "Speeding Up" or "Slowing Down" is the **relationship between the directions of $\vec{a}$ and $\vec{v}$**.

| Sign of $v$ (Direction of Motion) | Sign of $a$ (Direction of Force) | Relationship | Motion State (Speed Change) |

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

| $+$ (Right) | $+$ (Right) | **Same** | **Speeding Up** |

| $+$ (Right) | $-$ (Left) | **Opposite** | **Slowing Down** |

| $-$ (Left) | $-$ (Left) | **Same** | **Speeding Up** |

| $-$ (Left) | $+$ (Right) | **Opposite** | **Slowing Down** |

**Summary**: Same signs = Speed up; Opposite signs = Slow down.

3. Graphical Analysis ($v-t$ Graph)

In a Velocity-Time graph, acceleration is represented by geometric features.

* **Slope**: The slope of the tangent line in a $v-t$ graph represents **Instantaneous Acceleration**.

* Positive Constant Slope $\to$ Uniform Acceleration.

* Zero Slope $\to$ Constant Velocity.

* Changing Slope $\to$ Variable Acceleration.

* **Area**: The area under an Acceleration-Time ($a-t$) graph represents the **Change in Velocity ($\Delta v$)**.

4. Typical Examples

**Example 1: Does Negative Acceleration Mean Slowing Down?**

**Question**: An object moves West, and its acceleration is also directed West. Is its speed increasing or decreasing?

**Analysis**:

1. Define East as Positive (+), West as Negative (-).

2. Moves West $\to v < 0$.

3. Acceleration West $\to a < 0$.

4. $a$ and $v$ have the **same sign**, so the object is **Speeding Up**.

**Example 2: The Braking Problem**

**Question**: A car travels at $20\text{m/s}$ and brakes with an acceleration magnitude of $5\text{m/s}^2$. Find the displacement during the **last** $2\text{s}$ before it stops.

**Trick**: Don't calculate the first 2 seconds of braking. The question asks for the "last" 2 seconds.

**Technique**:

* **Reverse Thinking**: Treat the braking process (decelerating to 0) in reverse as accelerating from 0.

* $x = \frac{1}{2}at^2 = \frac{1}{2}(5)(2)^2 = 10\text{m}$.

* This utilizes symmetry and avoids calculating exactly when the final 2s interval starts.