Q.$A$ runs 3 times as fast as $B$. If $A$ gives $B$ a start of 40 metres, how far must the goal on the race course be so that $A$ and $B$ reach the goal at the same time?

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 …