Motion & Time
Speed as a calculated quantity, and reading motion from a graph.
Describing motion precisely
An object is in motion if its position changes relative to something else.
Relative to what matters. Sitting on a moving bus you are still relative to the bus, and moving at 60 km/h relative to the road. Neither is wrong, motion is always measured against a chosen reference.
To describe it quantitatively we need distance and time.
The speed formula
Rearranged as needed:
distance = speed × timetime = distance ÷ speed
Units must match. Metres and seconds give m/s; kilometres and hours give km/h. Mixing them gives nonsense.
Converting between the two:
to go km/h → m/s, divide by 3.672 km/h = 20 m/s
Average speed is not instant speed
Average speed is total distance divided by total time. It says nothing about what happened in between.
A car travelling 120 km in 2 hours has an average speed of 60 km/h, even if it stopped for lunch and then drove at 100.
Instantaneous speed is the speed at one moment, which is what a speedometer shows.
Why the distinction matters: a speeding fine is based on instantaneous speed, but average-speed cameras measure the other kind, which is why slowing down only at the camera does not work.
Reading a distance-time graph
A distance-time graph tells you everything about a journey, if you know how to read the gradient.
- Straight sloping line: steady speed.
- Steeper line: faster.
- Horizontal line: stationary. Time passes, distance does not change.
- Curve getting steeper: accelerating.
The gradient IS the speed:
gradient = change in distance ÷ change in time= speed
The most common error: reading a flat line as "not moving in time" or as slow movement. Flat means stopped. A line going back down to zero would mean returning to the start.
A worked journey
Question. A cyclist rides 12 km in 40 minutes, rests 20 minutes, then rides 8 km in 30 minutes. Find the average speed for the whole journey in km/h.
Step 1: total distance.
12 + 8 = 20 kmStep 2: total time, including the rest. This is the step people miss.
40 + 20 + 30 = 90 min = 1.5 hoursStep 3: divide.
20 ÷ 1.5 = 13.3 km/hWhy the rest counts: average speed is total distance over total elapsed time. Excluding the rest would give the average speed while moving, a different quantity, and not what was asked.
Speed, velocity and acceleration
Speed is how fast, a number with a unit.
Velocity is speed with a direction. 20 m/s north.
Why the difference matters: a car going round a roundabout at a constant 30 km/h has constant speed but changing velocity, because the direction keeps changing.
Acceleration is the rate at which velocity changes.
acceleration = change in velocity ÷ timeSo the car on the roundabout is accelerating even at steady speed, which sounds wrong until you accept that changing direction is a change in velocity.
Slowing down is negative acceleration, often called deceleration.
Motion in daily life
Road safety. Stopping distance has two parts: thinking distance (before you react) and braking distance. Braking distance grows with the square of speed, doubling your speed roughly quadruples it. This is the main argument for speed limits.
Journey planning. Distance ÷ speed gives an estimate; real journeys need traffic and stops added.
Sport. Race times, run rates, and analysing a bowler's delivery speed.
Space. Light takes about 8 minutes to reach us from the Sun, so you always see it as it was 8 minutes ago.
Where it usually goes wrong
- Flat line on a distance-time graph means slow. It means stopped.
- Excluding rest time from average speed.
- Mixing units: km with seconds.
- "Constant speed means no acceleration." Not if direction changes.
- "Speed and velocity are the same." Velocity has direction.
What to remember
- speed = distance ÷ time; keep units consistent.
- Average uses total distance and total time.
- On a distance-time graph the gradient is the speed; flat = stopped.
- Velocity has direction; acceleration is change of velocity.
Watch one worked through
Here is one done slowly, with the reason for every move. Then one for you.
A bus travels 180 km in 3 hours. Find its average speed, then find how far it goes in 5 hours at that speed.
- Write the formula and the values.speed = distance ÷ time = 180 ÷ 3Writing the formula first earns a mark on its own and stops you dividing the wrong way round.
- Divide to get the speed.180 ÷ 3 = 60 km/hThe unit comes from the formula: kilometres divided by hours gives km/h. The unit is part of the answer.
- For the second part rearrange the same formula.distance = speed × time = 60 × 5One formula gives all three quantities. Rearranging it beats memorising three separate ones.
- Multiply.60 × 5 = 300 kmAverage speed does not mean it travelled at 60 the whole way. It means the same total distance in the same total time.
Answer: 60 km/h, and 300 km in 5 hours
A car travels 150 km in 2 hours. What is its average speed in km/h?
Show the answer
75
Practise Motion & Time
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