Q.In a 1000 metres race, $A$, $B$ and $C$ get the gold, silver and bronze medals, respectively. If $A$ beats $B$ by 100 metres and $B$ beats $C$ by 100 metres, then by how many metres does $A$ beat $C$?

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.

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 $v_{AB}$ be the velocity of A relative to B, and $v_{CB}$ be the velocity of C relative to B. Then the relative speed ratio of A to C with respect to B is:

where $|v|$ denotes speed (magnitude of velocity).

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.

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.

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.