Relative Speed: Formula, Rules and Solved Questions

Relative Speed measures how fast one object appears to move with respect to another object. Its value depends on the directions of motion: speeds are subtracted when objects move in the same direction and added when they move in opposite directions. This page covers the relative speed formula, relative motion rules and calculation methods.

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What Is Relative Speed?

Relative speed is the rate at which the distance between two moving objects changes with respect to each other. For one-dimensional motion, it is the difference of speeds in the same direction and the sum of speeds in opposite directions.

If two objects move in the same direction with speeds v₁ and v₂, their relative speed is |v₁ − v₂|. If they move in opposite directions, their relative speed is v₁ + v₂. The time taken to cover a relative distance is calculated as time = relative distance ÷ relative speed.

Relative Speed Formula & Tricks

Important Formulas

Same-direction relative speed
Relative speed = |v₁ − v₂|

Subtract the smaller speed from the greater speed when both objects move in the same direction.

Opposite-direction relative speed
Relative speed = v₁ + v₂

Add the speeds when the objects move towards each other or away from each other in opposite directions.

Relative motion time
Time = Relative distance ÷ Relative speed

Use the initial gap, length or separation as the relative distance, with consistent units.

Catch-up time
Catch-up time = Initial gap ÷ (Faster speed − Slower speed)

This applies when two objects move in the same direction and the faster object is behind the slower object.

Quick Tricks

Use subtraction for the same direction

When two objects travel in the same direction, the distance between them changes at the difference of their speeds.

Example: A car at 72 km/h overtakes a car at 54 km/h. Relative speed = 72 − 54 = 18 km/h.
Use addition for opposite directions

When two objects move towards each other, both cover part of the gap, so their speeds are added.

Example: Two cyclists move towards each other at 20 km/h and 25 km/h. Relative speed = 20 + 25 = 45 km/h.
Convert units before calculating

Use 1 km/h = 5/18 m/s and 1 m/s = 18/5 km/h. Distance and speed must use compatible units.

Example: A relative speed of 54 km/h equals 54 × 5/18 = 15 m/s.

Relative Speed Concepts

Relative Speed in the Same Direction

For objects moving in the same direction, relative speed is the absolute difference between their speeds.

If the speeds are v₁ and v₂, then relative speed = |v₁ − v₂|. The faster object closes a gap only when it is behind the slower object. If both speeds are equal, the relative speed is zero and the distance between them remains constant.

Example: A bus travels at 60 km/h and a car travels at 80 km/h in the same direction. Relative speed = 80 − 60 = 20 km/h.

Relative Speed in Opposite Directions

For objects moving in opposite directions, relative speed is the sum of their speeds.

When two objects approach each other, the distance between them decreases at v₁ + v₂. The same addition applies when they move away from each other in opposite directions. If the initial distance is d, time to meet or separate by d is d ÷ (v₁ + v₂).

Example: Two trains are 180 km apart and move towards each other at 70 km/h and 50 km/h. Time to meet = 180 ÷ 120 = 1.5 hours.

Catch-Up and Overtaking Problems

In a catch-up problem, time equals the initial gap divided by the difference in speeds.

For a faster object moving behind a slower object in the same direction, catch-up time = initial gap ÷ (faster speed − slower speed). The faster object must have a greater speed; otherwise, it cannot close the gap.

Example: A runner is 100 m behind another runner. Their speeds are 8 m/s and 6 m/s. Catch-up time = 100 ÷ (8 − 6) = 50 seconds.

Relative Speed of Trains

For trains, use the train lengths as the distance to be covered during crossing or overtaking.

When two trains cross in opposite directions, distance covered relative to each other is the sum of their lengths. When they cross in the same direction, it is the sum of their lengths and relative speed is the difference of their speeds. Convert km/h to m/s when lengths are given in metres.

Example: Two trains of lengths 120 m and 180 m move in opposite directions at 36 km/h and 54 km/h. Relative speed = 90 km/h = 25 m/s. Crossing time = (120 + 180) ÷ 25 = 12 seconds.

Relative Speed Video Lessons

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Relative Speed in Opposite Directions

Learn how to calculate relative speed when two objects move in opposite directions, and apply the method to time, speed and distance problems.

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Practice Relative Speed Questions

Practise published questions related to this topic.

1What is the largest four-digit number that is completely divisible by 18, 30 and 45 ?→ 2Two numbers have HCF 14 and LCM 420. If their difference is 98, find the numbers.→ 3An article is sold for 25 percent loss. The value of discount given is equal to (1/3) of the marked price. What is the marked price of the article?→ 4Which single digit should replace the blank in the number 1_36 so that the number is divisible by 9?→ 5A trader sells an item at a 25% loss. If he had sold it for ₹60 more, he would have made a 10% profit. Find the cost price.→ 6A shopkeeper offers the following schemes on toys of the same marked price. (A) Successive discounts of 27% and 39% on any number of toys bought. (B) Successive discounts of 39%, 7% and 40% on any number of toys bought. (C) 26% discount on the first 9 toys and 13% discount on each toy thereon. (D) 7 toys free of cost on buying 9 toys. A customer wants to buy 9 toys. Which of the above schemes is the least beneficial to her?→ 7A person sells an article at the profit of 50 percent. Cost price and selling price are increased by 40 percent and 'y' percent respectively. If the new profit percentage is 75 percent, then what is the value of 'y'?→ 8​Identify the number that is divisible by both 2 and 3.→ 9A shopkeeper marks his goods at X% above the cost price and sells them at a discount of 35%. If he makes a profit of 69%, find the value of X.→ 10Arjun bought some erasers at the rate of ₹180 a dozen. He sold them for ₹26 each. His profit percentage is ______% (rounded off to two decimals).→

Relative Speed Quick Quiz

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Quick Revision Notes

Relative Speed Revision Points

Use these rules to solve relative motion calculations.

  • Same direction: relative speed = |v₁ − v₂|.
  • Opposite directions: relative speed = v₁ + v₂.
  • Time = relative distance ÷ relative speed.
  • Catch-up time = initial gap ÷ difference in speeds.
  • For train crossing, relative distance is usually the sum of the train lengths.
  • Use 1 km/h = 5/18 m/s and 1 m/s = 18/5 km/h.
  • If two objects have equal speeds in the same direction, their relative speed is zero.

Relative Speed FAQs

What is the relative speed formula for two objects moving in the same direction?

Relative speed = |v₁ − v₂|. For speeds of 75 km/h and 55 km/h, the relative speed is 20 km/h.

What is the relative speed formula when two objects move in opposite directions?

Relative speed = v₁ + v₂. For speeds of 40 km/h and 35 km/h, it is 75 km/h.

How is catch-up time calculated?

Catch-up time = initial gap ÷ difference in speeds. A gap of 150 m with a speed difference of 5 m/s takes 150 ÷ 5 = 30 seconds.

What is the relative speed of two objects moving at equal speeds in the same direction?

Their relative speed is zero because |v − v| = 0. Therefore, the distance between them does not change.

How do you find the time for two objects moving towards each other?

Time = initial distance ÷ sum of speeds. If the distance is 240 km and speeds are 80 km/h and 40 km/h, time = 240 ÷ 120 = 2 hours.

What distance is used when two trains cross each other?

The relative distance is the sum of their lengths. For trains of lengths 100 m and 150 m, the distance to be covered is 250 m.

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