In physics, you find speed from velocity by determining the magnitude of the velocity vector.
Speed is essentially the "how fast" part of velocity, without the direction.
Understanding Speed and Velocity
Let's start by understanding the definitions provided:
- Velocity (v): Velocity is a vector quantity. This means it has both magnitude (speed) and direction. It measures the change in position (displacement, Δs) over the change in time (Δt), represented by the equation:
v = Δs / Δt
Displacement is the straight-line distance between the starting and ending points, including direction. - Speed (r): Speed is a scalar quantity. This means it only has magnitude; it does not include direction. It measures the total distance traveled (d) over the change in time (Δt), represented by the equation:
r = d / Δt
Distance is the total path length covered.
The Relationship: Speed is the Magnitude of Velocity
The speed at any instant (instantaneous speed) is the magnitude of the velocity vector at that same instant (instantaneous velocity).
Think of it this way: if an object's velocity is described as "10 meters per second North," its speed is simply "10 meters per second." The "North" part gives the direction, which is included in velocity but excluded for speed.
Key Takeaway: To get speed from velocity, you take the absolute value (or magnitude) of the velocity.
Practical Examples
Here are a few scenarios illustrating the relationship:
- Straight-Line Motion without Changing Direction: If a car travels 50 km East in a straight line for 1 hour, its displacement (Δs) is 50 km East, and the distance traveled (d) is also 50 km.
- Velocity = 50 km East / 1 hour = 50 km/h East
- Speed = 50 km / 1 hour = 50 km/h
- In this case, the magnitude of velocity (50 km/h) equals the speed (50 km/h).
- Straight-Line Motion Changing Direction: If a car travels 50 km East and then turns around and travels 50 km West, both in 1 hour each (total 2 hours):
- For the first hour: Velocity = 50 km/h East, Speed = 50 km/h
- For the second hour: Velocity = 50 km/h West, Speed = 50 km/h
- Over the entire 2 hours:
- Total Distance (d) = 50 km + 50 km = 100 km
- Total Displacement (Δs) = 50 km East + 50 km West = 0 km (assuming it ends where it started relative to the original point after the first leg)
- Average Speed = 100 km / 2 hours = 50 km/h
- Average Velocity = 0 km / 2 hours = 0 km/h
- Here, the average speed (50 km/h) is not the magnitude of the average velocity (0 km/h). This highlights that speed is related to distance and velocity to displacement. However, at any instant, the instantaneous speed is the magnitude of the instantaneous velocity. If the velocity is -50 km/h (meaning 50 km/h in the negative direction), the speed is | -50 km/h | = 50 km/h.
- Motion Along a Curved Path: If a runner runs a 400-meter lap on a track in 50 seconds:
- Total Distance (d) = 400 meters
- Total Displacement (Δs) = 0 meters (since they end where they started)
- Average Speed = 400 m / 50 s = 8 m/s
- Average Velocity = 0 m / 50 s = 0 m/s
- Again, the average speed is not the magnitude of the average velocity. However, at any point on the track, the runner's instantaneous speed is the magnitude of their instantaneous velocity vector (which is tangent to the track at that point).
Summary Table
| Feature | Speed (r) | Velocity (v) |
|---|---|---|
| Type | Scalar Quantity (Magnitude only) | Vector Quantity (Magnitude & Direction) |
| Measures | Distance traveled (d) over time | Displacement (Δs) over time |
| Equation | r = d / Δt |
v = Δs / Δt |
| Relation | Magnitude of velocity | Includes speed as its magnitude |
In essence, if you have a velocity value, its magnitude is the speed. For example, a velocity of (3 m/s East, 4 m/s North) has an instantaneous speed calculated using the Pythagorean theorem for magnitude: sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 m/s. The speed is 5 m/s.
