3.3 Speed, Distance, Time & Rate Calculations
Key Takeaways
- The fundamental relationship between distance, speed, and time is expressed as D = S × T, S = D / T, and T = D / S.
- Always ensure your units are consistent before performing calculations (e.g., if speed is in km/h, time must be in hours).
- To convert minutes to hours, divide by 60; to convert hours to minutes, multiply by 60.
- When dealing with relative speed, add speeds if objects move in opposite directions and subtract if moving in the same direction.
Speed, Distance, Time & Rate Calculations
Calculations involving speed, distance, and time are incredibly common in the Defence Aptitude Assessment. Whether it's calculating the ETA of a convoy or the distance covered by an aircraft, mastering these relationships is a necessity.
The DST Triangle
The relationship between distance, speed, and time can be easily remembered using the DST triangle, where Distance (D) is at the top, and Speed (S) and Time (T) are at the bottom.
From this triangle, we can derive three formulas:
- Distance = Speed × Time
- Speed = Distance / Time
- Time = Distance / Speed
Calculating Speed
Speed is the rate at which an object covers distance.
Worked Example: A naval vessel travels 135 nautical miles in 4.5 hours. What is its average speed in knots (nautical miles per hour)?
- Identify the formula: Speed = Distance / Time.
- Plug in the values: Speed = 135 / 4.5.
- Calculation trick: Multiply both by 2 to remove the decimal: 270 / 9 = 30.
- The speed is 30 knots.
Calculating Distance
Distance is a measure of how much ground an object has covered during its motion.
Worked Example: A helicopter flies at an average speed of 120 km/h for 2 hours and 15 minutes. How far does it travel?
- First, ensure units match. Time is given in hours and minutes. Convert 15 minutes to hours: 15 / 60 = 0.25 hours.
- Total time = 2.25 hours.
- Identify the formula: Distance = Speed × Time.
- Calculate: 120 × 2.25. (120 × 2 = 240, and 120 × 0.25 = 30. 240 + 30 = 270).
- The distance traveled is 270 km.
Calculating Time
Time is the duration in which an object covers a certain distance at a given speed.
Worked Example: A convoy needs to travel 360 km at an average speed of 80 km/h. How long will the journey take in hours and minutes?
- Identify the formula: Time = Distance / Speed.
- Calculate: 360 / 80 = 4.5 hours.
- Convert the decimal part back to minutes: 0.5 hours × 60 = 30 minutes.
- The journey takes 4 hours and 30 minutes.
Consistent Units and Conversions
The most common mistake candidates make is failing to align their units. If your speed is in meters per second (m/s), your distance must be in meters, and your time in seconds.
Common Conversions
| From | To | Conversion Factor |
|---|---|---|
| Hours | Minutes | Multiply by 60 |
| Minutes | Hours | Divide by 60 |
| km/h | m/s | Divide by 3.6 |
| m/s | km/h | Multiply by 3.6 |
Worked Example: Convert 72 km/h to m/s. Calculation: 72 / 3.6 = 20 m/s. (This is because 72 km = 72,000 meters, and 1 hour = 3,600 seconds. 72,000 / 3,600 = 20).
Relative Speed
Relative speed problems involve two moving objects. You must determine how fast the distance between them is changing.
Moving in Opposite Directions
When two objects move towards each other, or move away from each other from a single point, their relative speed is the sum of their individual speeds.
Worked Example: Two aircraft are 900 km apart and fly directly towards each other. Aircraft A flies at 400 km/h and Aircraft B flies at 500 km/h. How long until they intercept?
- Calculate relative speed: 400 + 500 = 900 km/h.
- Use the time formula: Time = Distance / Relative Speed.
- Time = 900 / 900 = 1 hour.
Moving in the Same Direction
When two objects travel in the same direction, the relative speed is the difference between their speeds. This is used when one object is catching up to another.
Worked Example: A suspect vehicle is fleeing at 120 km/h. A pursuit vehicle is 10 km behind, traveling at 140 km/h. How long will it take the pursuit vehicle to catch the suspect?
- Calculate relative speed: 140 - 120 = 20 km/h. (The pursuit vehicle closes the gap by 20 km every hour).
- Use the time formula: Time = Distance / Relative Speed.
- Time = 10 / 20 = 0.5 hours.
- Convert to minutes: 0.5 × 60 = 30 minutes.
Rate of Work
Work rate problems are mathematically similar to speed problems. Instead of distance, you are dealing with a task to be completed.
Worked Example: Pump A can fill a tank in 4 hours. Pump B can fill the same tank in 6 hours. How long will it take if they work together?
- Find the rate per hour for each pump: Pump A does 1/4 of the job per hour. Pump B does 1/6 of the job per hour.
- Combine their rates: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 of the job per hour.
- The time taken is the reciprocal of the combined rate: 12/5 hours.
- Convert to hours and minutes: 12/5 = 2.4 hours. 0.4 hours × 60 = 24 minutes. Total time is 2 hours and 24 minutes.
An armored personnel carrier travels at a constant speed of 45 km/h for 40 minutes. How far does it travel?
Two patrol boats depart from the same port simultaneously but travel in completely opposite directions. Boat Alpha travels at 25 knots and Boat Bravo travels at 35 knots. How long will it take for them to be 150 nautical miles apart?
A fast jet needs to cover a distance of 1,200 km in exactly 1 hour and 15 minutes. What must its average speed be?