Speed, Distance and Time (with Time and Work)

Speed, distance and time

Speed tells how much distance is covered in one unit of time.

Speed = Distance ÷ Time   Distance = Speed × Time   Time = Distance ÷ Speed

SI unit: metre per second (m/s). On roads we use km/h.

• km/h → m/s: multiply by 1000/3600 = 5/18. So 72 km/h = 20 m/s.
• m/s → km/h: multiply by 18/5. So 15 m/s = 54 km/h.

Average speed = total distance ÷ total time. It is NOT the average of the speeds. Going 60 km at 30 km/h and coming back at 60 km/h: time = 2 h + 1 h = 3 h, so average speed = 120 ÷ 3 = 40 km/h (not 45). For equal distances, average speed = 2xy ÷ (x + y).

Distance–time graphs

Put time on the x-axis and distance on the y-axis.

• A straight sloping line = constant speed. Slope = speed (rise ÷ run).
• A steeper line = faster.
• A flat (horizontal) line = stopped.
• A curve getting steeper = speeding up.
• Where two lines cross, the two bodies are at the same place at the same time (they meet).

Relative speed and trains

When two bodies move:

• Towards each other (opposite directions): relative speed = a + b.
• Same direction: relative speed = a − b (the faster one gains on the slower).

Time to meet or catch up = gap ÷ relative speed.

Trains: a train has length, so it is not a point.

• Passing a pole or a person: distance = length of train.
• Passing a platform or a bridge: distance = train + platform.
• Two trains passing each other: distance = sum of both lengths, speed = relative speed.

Boats and streams; races

Let the boat's speed in still water be u and the stream's speed be v.

• Downstream (with the flow): u + v
• Upstream (against the flow): u − v
• So u = (down + up) ÷ 2 and v = (down − up) ÷ 2.

Races: "A beats B by 20 m" in a 100 m race means when A finishes 100 m, B has run 80 m. "A gives B a start of 10 m" means B starts 10 m ahead. In equal times, distances are in the ratio of speeds.

Time and work; pipes and cisterns

If a person finishes a job in n days, their work rate is 1/n of the job per day. Rates add when people work together.

A and B finish in 12 and 6 days: together 1/12 + 1/6 = 3/12 = 1/4 per day, so 4 days.

Work and time are inversely proportional for a fixed job: twice the workers → half the time. Graph of time against number of workers is a falling curve (a hyperbola); graph of work done against time at a fixed rate is a straight line through the origin.

Pipes and cisterns: a filling pipe adds its rate (+), an emptying pipe or leak takes its rate away (−). Tank filled by a pipe in 4 h but a leak empties it in 12 h: net rate 1/4 − 1/12 = 1/6, so 6 h.

Try it at home

Mark 20 m in a park and time yourself walking and running. Work out your speed in m/s and km/h. Then fill a bucket with one tap, and again with two taps, and check whether the times match the rate rule.

Key formulas and definitions

• Speed = Distance ÷ Time
• x km/h = x × 5/18 m/s; y m/s = y × 18/5 km/h
• Average speed = total distance ÷ total time (equal distances: 2xy/(x+y))
• Relative speed: opposite = a + b, same direction = a − b
• Train passing platform: time = (train + platform length) ÷ speed
• Downstream = u + v; upstream = u − v
• Work rate = 1/n per day; together: 1/a + 1/b; leak: subtract

Worked examples

Common mistakes

• Taking average speed as (x + y) ÷ 2. Always use total distance ÷ total time.
• Mixing units: speed in km/h but length in metres. Convert first.
• Forgetting the train's own length when it passes a platform or another train.
• Adding speeds for the same direction. Same direction → subtract; opposite → add.