Speed: The Rate of Change of Distance
Defining Speed and Its Core Principles
At its simplest, speed tells us how fast something is moving. The scientific definition is the distance an object travels per unit of time. Since it only has magnitude (a numerical value) and no direction, it is classified as a scalar quantity. This is a key point. Saying "a car moves at 60 km/h" is a statement of speed; we don't know if it's going north, south, east, or west.
The standard unit for speed in the International System of Units[1] (SI) is metres per second (m/s). This means if an object has a speed of 5 m/s, it covers a distance of 5 metres every second. Another common unit is kilometers per hour (km/h).
$$ \text{Speed} = \frac{\text{Distance Traveled}}{\text{Time Taken}} $$
This can be written symbolically as: $$ v = \frac{d}{t} $$ where \( v \) represents speed, \( d \) represents distance, and \( t \) represents time.
Example: If a cyclist rides a distance of 150 metres in a time of 30 seconds, their speed is calculated as:
$$ v = \frac{d}{t} = \frac{150 \text{ m}}{30 \text{ s}} = 5 \text{ m/s} $$
This means the cyclist covers 5 metres every second.
Average Speed vs. Instantaneous Speed
It is rare for any object to maintain a constant speed for an entire journey. Cars stop at traffic lights, and runners start slow and then speed up. This is why we have two important concepts: average speed and instantaneous speed.
| Type of Speed | Definition | Example |
|---|---|---|
| Average Speed | The total distance traveled divided by the total time taken for the entire journey. | A car trip from city A to city B covers 120 km in 2 hours. The average speed is \( \frac{120}{2} = 60 \text{ km/h} \). This doesn't mean the car was always at 60 km/h. |
| Instantaneous Speed | The speed of an object at a specific moment in time. | Looking at a car's speedometer and seeing 85 km/h is reading its instantaneous speed at that exact moment. |
The Crucial Difference: Speed vs. Velocity
This is one of the most important distinctions in physics. While speed is a scalar (magnitude only), velocity is a vector (magnitude and direction).
Example to Illustrate: Imagine a car moving in a circle at a steady rate. Its speed might be constant, say 50 km/h. However, its velocity is constantly changing because the direction of travel is continuously changing. If you finish a lap and return to your starting point, your average velocity for the entire lap is zero because your total displacement[2] is zero, even though your average speed was 50 km/h.
| Aspect | Speed | Velocity |
|---|---|---|
| Quantity Type | Scalar | Vector |
| What it Answers | How fast? | How fast and in which direction? |
| Can it be Zero? | Yes, when the object is stationary (no distance covered). | Yes, when the object is stationary OR when it returns to its start point (zero displacement). |
Calculating Speed in Real-World Scenarios
Let's apply the speed formula to more complex, real-life situations. This often requires careful consideration of the total distance and total time.
Example 1: The Journey with a Stop
A student walks from home to school, a distance of 1.5 km, in 25 minutes. After school, they walk back home along the same path, but this time it takes 35 minutes. What is the student's average speed for the entire round trip?
Step 1: Find the Total Distance. Round trip means going to school and back. So, total distance \( d_{total} = 1.5 \text{ km} + 1.5 \text{ km} = 3.0 \text{ km} \).
Step 2: Find the Total Time. Total time \( t_{total} = 25 \text{ min} + 35 \text{ min} = 60 \text{ min} \). For the formula, we need consistent units. Let's convert 60 minutes to hours: \( 60 \text{ min} = 1 \text{ hour} \).
Step 3: Apply the Formula.
$$ v_{average} = \frac{d_{total}}{t_{total}} = \frac{3.0 \text{ km}}{1 \text{ h}} = 3.0 \text{ km/h} $$
The student's average speed for the entire journey is 3.0 km/h.
Example 2: Converting Units
It is very common to need to convert between m/s and km/h. The conversion factor is:
$$ 1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{5}{18} \text{ m/s} $$
So, to convert from km/h to m/s, you multiply by \( \frac{5}{18} \).
If a train is moving at 108 km/h, what is its speed in m/s?
$$ v = 108 \text{ km/h} \times \frac{5}{18} = 30 \text{ m/s} $$
The train's speed is 30 metres per second.
Common Mistakes and Important Questions
Q: Is it possible for an object to have a constant speed but a changing velocity?
A: Yes, absolutely. This happens when the object is moving at a steady rate (constant speed) but its direction is changing. The classic example is an object moving in a uniform circular path.
Q: When calculating average speed, why do we use total distance and not net displacement?
A: Because speed is a scalar quantity related to the path covered, regardless of direction. Velocity uses displacement. If you run a 400 m lap on a track and finish where you started, your displacement is zero, so your average velocity is zero. However, you ran 400 m, so your average speed is a positive number calculated from that 400 m distance.
Q: What is the difference between "metres per second" (m/s) and "metres per second squared" (m/s²)?
A: Metres per second (m/s) is a unit of speed (or velocity). It tells you how many metres are traveled each second. Metres per second squared (m/s²) is a unit of acceleration. It tells you how much the speed (in m/s) changes every second. For example, an acceleration of 5 m/s² means the object's speed increases by 5 m/s with each passing second.
Footnote
[1] International System of Units (SI): The modern form of the metric system and the world's most widely used system of measurement. It establishes a standard set of units for all physical quantities, such as the metre for length and the second for time.
[2] Displacement: A vector quantity that refers to the change in an object's position. It is the straight-line distance from the initial point to the final point, with a direction. It is not necessarily the same as the total distance traveled.