Relative Speed Ratio — From Intuition to Precision
Imagine you're standing on a railway platform. A train passes you at 60 km/h. To you, it's moving fast. Now imagine you're sitting in another train running at 50 km/h in the same direction. The first train now seems to crawl past you at just 10 km/h. The same train, two different experiences — because your own motion changes how you perceive the other's speed.
That perceived speed is the relative speed. And when you compare two such relative speeds — say, the speed of one object relative to another, versus the speed of a third object relative to the same reference — you're dealing with a relative speed ratio.
Relative speed is always about one object as seen from another moving object. It is not the same as their individual speeds.
The Intuition First
Suppose two cars are on a straight road. Car A moves at 80 km/h, Car B at 60 km/h, both in the same direction. If you're in Car B, Car A approaches you at only 20 km/h. That's the relative speed: 80−60=20 km/h.
Now suppose Car C is coming from the opposite direction at 70 km/h. From Car B, Car C rushes toward you at 60+70=130 km/h.
The relative speed ratio compares two such relative speeds. For example, the ratio of Car A's speed relative to Car B, to Car C's speed relative to Car B, is 20:130=2:13.
That's all it is — a ratio of two relative speeds.
The Precise Statement
Let there be three objects (or two objects and a reference frame). Let vAB be the velocity of A relative to B, and vCB be the velocity of C relative to B. Then the relative speed ratio of A to C with respect to B is:
Relative Speed Ratio=∣vCB∣∣vAB∣
where ∣v∣ denotes speed (magnitude of velocity).
Relative Speed Ratio=speed of second object relative to same referencespeed of first object relative to reference
The reference object (B) is the one from whose perspective both speeds are measured.
Why It Matters
Relative speed ratios appear in:
- Time and distance problems — when two moving bodies meet or overtake, the ratio of their relative speeds determines the ratio of times or distances.
- River-boat problems — the ratio of the boat's speed relative to water to the river's speed relative to ground.
- Train-platform problems — comparing how fast two trains appear to cross a stationary observer.
In many exam problems, you don't need the absolute speeds — only the ratio of relative speeds. That ratio alone can give you the answer.
A Common Mistake
Students often confuse relative speed ratio with the ratio of the objects' own speeds. For example, if A moves at 80 km/h and C at 70 km/h, the ratio of their own speeds is 80:70=8:7. But if B is moving at 60 km/h in the same direction as A, the relative speed ratio is 20:130=2:13 — completely different.
Never substitute the objects' own speeds into a relative speed ratio. Always compute the relative speeds first.
One More Example
A train X is 200 m long and runs at 90 km/h. Train Y is 150 m long and runs at 72 km/h in the opposite direction. The relative speed of X with respect to Y is 90+72=162 km/h. The relative speed of Y with respect to X is also 162 km/h (same magnitude). So the relative speed ratio of X to Y (with respect to the ground) is 162:162=1:1 — they see each other at the same speed.
But if both ran in the same direction, the relative speed would be 90−72=18 km/h, and the ratio would be 18:18=1:1 again — because relative speed is symmetric in magnitude.
The ratio becomes interesting when you compare different pairs — like X relative to ground versus Y relative to ground, or X relative to Y versus Z relative to Y.
Final takeaway: Relative speed ratio is just a comparison of two apparent speeds from the same moving viewpoint. Compute each relative speed correctly, then take the ratio. That's the whole idea.