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Speed
Definition
Speed is the rate at which an object covers distance. It measures how fast an object is moving. Speed is a scalar quantity - it has only magnitude (no direction).Types of Speed
Average Speed
Average speed is the total distance traveled divided by the total time taken.\[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\]
\[v_{\text{avg}} = \frac{d}{\Delta t}\]
Where:
- d = total distance traveled
- Δt = total time elapsed
- Units: m/s (meters per second) or km/h (kilometers per hour)
- Always positive (distance is always positive)
- Does not show direction (scalar)
- Depends on the total path taken
- Useful for overall motion description
> [Ejemplo: A car travels 120 km in 2 hours. > Average speed = 120 km / 2 h = 60 km/h > > Even if the car traveled at different speeds during the journey (faster on highways, slower in cities), the average speed is 60 km/h.]
Instantaneous Speed
Instantaneous speed is the speed of an object at a specific moment in time.\[v = \lim_{\Delta t \to 0} \frac{\Delta d}{\Delta t}\]
Characteristics:
- Speed at a specific instant
- Can change from moment to moment
- The speedometer in a car shows instantaneous speed
- In calculus, it's the derivative of distance with respect to time
Ejemplo: Looking at a car's speedometer, it reads 60 km/h at a particular instant. This is the instantaneous speed. A moment later, it might read 65 km/h.
Relationship Between Average and Instantaneous Speed
- Average speed = overall rate for a journey
- Instantaneous speed = speed at any particular moment
- If an object moves at constant speed, average speed = instantaneous speed at any time
Units of Speed
Common units for speed:
| Unit | Symbol | Conversion |
|---|---|---|
| Meters per second | m/s | Base SI unit |
| Kilometers per hour | km/h | 1 km/h = 0.278 m/s |
| Miles per hour | mph | 1 mph = 0.447 m/s |
| Centimeters per second | cm/s | 1 cm/s = 0.01 m/s |
Converting Units
From km/h to m/s:\[\text{m/s} = \text{km/h} \times \frac{1000 \text{ m}}{3600 \text{ s}} = \text{km/h} \times \frac{1}{3.6}\]
> [Ejemplo: Convert 72 km/h to m/s: > 72 km/h ÷ 3.6 = 20 m/s]
From m/s to km/h:\[\text{km/h} = \text{m/s} \times 3.6\]
> [Ejemplo: Convert 10 m/s to km/h: > 10 m/s × 3.6 = 36 km/h]
Solving Speed Problems
Basic Speed Calculation
Problem: A runner completes a 100 m race in 10 seconds. What is the average speed?\[v_{\text{avg}} = \frac{100 \text{ m}}{10 \text{ s}} = 10 \text{ m/s}\]
Finding Distance
From the formula \(v = \frac{d}{t}\), we can rearrange to find distance:
\[d = v \times t\]
> [Ejemplo: A cyclist travels at an average speed of 15 m/s for 30 seconds. How far does the cyclist travel? > d = 15 m/s × 30 s = 450 m]
Finding Time
From the formula \(v = \frac{d}{t}\), we can rearrange to find time:
\[t = \frac{d}{v}\]
> [Ejemplo: A car travels 240 km at an average speed of 60 km/h. How long does the journey take? > t = 240 km / 60 km/h = 4 hours]
Speed vs Velocity
Although often used interchangeably, speed and velocity are different:
| Aspect | Speed | Velocity |
|---|---|---|
| Type | Scalar | Vector |
| Direction | Not included | Included |
| Definition | Rate of distance coverage | Rate of displacement |
| Formula | v = d/Δt | v = Δx/Δt |
| Example | "60 km/h" | "60 km/h north" |
| Can be zero | Only if no motion | Yes, if return to start |
Applications of Speed
1. Transportation
- Speed limits on roads
- Average speeds for trip planning
- Speedometers in vehicles
2. Astronomy
- Speed of light: c = 3 × 10⁸ m/s
- Speeds of planets orbiting the Sun
- Speeds of galaxies
3. Sports
- Speed of athletes in races
- Speed of balls in sports
- Average speeds in competitions
4. Physics Experiments
- Measuring particle speeds
- Analyzing collision speeds
- Studying wave propagation
Important Speed Values to Remember
| Object | Speed |
|---|---|
| Walking | 1-2 m/s |
| Running | 3-6 m/s |
| Cycling | 5-10 m/s |
| Car on highway | 25-35 m/s |
| Airplane | 200-250 m/s |
| Sound in air | 343 m/s (at 20°C) |
| Light in vacuum | 3 × 10⁸ m/s |