Relative Velocity in 1D

The motion of one object as seen from another is governed by relative velocity. In 1D, this is straightforward arithmetic — but careful with signs.

Concept

If $A$ has velocity $v_A$ and $B$ has velocity $v_B$ (both measured in a common frame), then the velocity of $A$ relative to $B$ is:

$v_{AB} = v_A - v_B.$

Similarly $v_{BA} = v_B - v_A = -v_{AB}$.

Special Cases

• Same direction: Use the sign convention; $v_{AB}$ has small magnitude (the difference).
• Opposite directions: One velocity is negative; $v_{AB}$ has large magnitude (their sum).

Closing/Separation Speed

When two bodies approach each other along a line: $\text{closing speed} = |v_A - v_B| \text{ (if same sign of approach)}$ Pay attention to which direction each is moving.

Worked Examples

Example 1: Trains on Parallel Tracks

Q: Train A moves at 50 m/s east; train B moves at 30 m/s east. Find $v_{AB}$ (velocity of A as seen by B) and $v_{BA}$.

Solution: With east positive: $v_{AB} = 50 - 30 = 20$ m/s east. $v_{BA} = -20$ m/s east = 20 m/s west.

Example 2: Head-on Approach

Q: Two trains approach each other on the same track. Train A at 40 m/s east; train B at 30 m/s west. Initial separation 700 m. When do they meet?

Solution: Take east positive. $v_A = +40$, $v_B = -30$. Closing speed $= v_A - v_B = 40 - (-30) = 70$ m/s.

Time to meet $= 700 / 70 = 10$ s.

Example 3: Overtake Problem

Q: A car at 25 m/s overtakes a truck moving at 15 m/s in the same direction. The car is initially 100 m behind. How long to overtake (front-to-front)?

Solution: Relative velocity of car w.r.t. truck: $25 - 15 = 10$ m/s.

Time to close 100 m gap: $100/10 = 10$ s.

Example 4: Time of Crossing

Q: A 200-m-long train moves at 20 m/s east; a 100-m-long train moves at 10 m/s west on a parallel track. How long do they take to pass each other completely?

Solution: Relative speed (head-on): $20 + 10 = 30$ m/s. Total length to clear: $200 + 100 = 300$ m. Time = $300/30 = 10$ s.

Common Confusions

• "Relative velocity is just the sum of velocities." — Only when motion is in opposite directions. Always use $v_A - v_B$ with sign conventions; the "sum" feel emerges when one $v$ is negative.
• "Relative velocity depends on frames." — Yes — what matters is the difference between two velocities in the same frame.
• A car at rest in your frame may be moving fast relative to a third observer.

Key Takeaways

• $v_{AB} = v_A - v_B$ in 1D with consistent signs.
• Same-direction motion: small relative speed (difference).
• Opposite-direction motion: large relative speed (effective sum).
• For crossing problems: relative speed × time = relative displacement (sum of lengths).