Speed and Velocity

Physics

1. Fundamental Concepts

Definition: Speed is the rate at which an object covers distance, calculated as $$\text{speed} = \frac{\text{distance}}{\text{time}}$$
Velocity: Velocity is a vector quantity that includes both speed and direction, calculated as $$\text{velocity} = \frac{\Delta \text{position}}{\Delta \text{time}}$$
Scalar vs Vector: Speed is a scalar quantity (magnitude only), while velocity is a vector quantity (magnitude and direction).

2. Key Concepts

Average Speed: $$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$$

Instantaneous Speed: The speed at a particular instant in time.

Average Velocity: $$\text{average velocity} = \frac{\Delta \text{position}}{\Delta \text{time}}$$

3. Examples

Example 1 (Basic)

Problem: A car travels 100 meters in 5 seconds. Calculate its average speed.

Step-by-Step Solution:

Use the formula for average speed: $$\text{average speed} = \frac{\text{distance}}{\text{time}}$$
Substitute the given values: $$\text{average speed} = \frac{100 \text{ meters}}{5 \text{ seconds}} = 20 \text{ m/s}$$

Validation: The calculation confirms that the car's average speed is 20 meters per second.

Example 2 (Intermediate)

Problem: A runner completes a 400-meter track in 50 seconds. What is the runner's average velocity if they start and finish at the same point?

Step-by-Step Solution:

Since the runner starts and finishes at the same point, the displacement is zero: $$\Delta \text{position} = 0$$
Calculate the average velocity using the formula: $$\text{average velocity} = \frac{\Delta \text{position}}{\Delta \text{time}} = \frac{0}{50 \text{ seconds}} = 0 \text{ m/s}$$

Validation: The runner's average velocity is 0 m/s because there is no net displacement.

4. Problem-Solving Techniques

Identify Variables: Clearly define all variables before solving problems.
Use Diagrams: Draw diagrams to visualize the motion and understand the direction of vectors.
Check Units: Ensure that all units are consistent throughout the calculations.