Distance-time models
Here distance means position from a reference point on one side of a straight route. Constant-speed segments and no unshown reversals make distance changes sufficient to calculate travel
- Read the quantity Distance from the reference point can fall; cumulative distance travelled cannot
- Use a segment Subtract both endpoint times and distances. A negative signed rate means movement towards the reference point
- Total and average Add absolute changes for total travel, then divide by full elapsed time for average speed
- Build a graph Choose units, calculate successive endpoints, join in time order and verify rests and returns
Worked example
A walker follows a straight route. Time is in hours and distance from home in kilometres: (0,0), (2,8), (3,8), (5,2). Find return speed, total distance and average speed
- \(\displaystyle v_{\mathrm{return}}=\frac{|2-8|}{5-3}=3\text{ km/h}\)Use only the return segment; speed is positive
- \(\displaystyle D=8+0+6=14\text{ km}\)Rest adds zero. Returning adds 6 km
- \(\displaystyle v_{\mathrm{average}}=14/5=2.8\text{ km/h}\)The full 5 hours include the one-hour rest
Common mistakes
Difficulty levels
Questions about this topic
Does a falling graph mean negative speed?+
No. On a distance-from-reference graph it means movement towards the reference point. Speed is the magnitude of the signed gradient
Can a cumulative-distance graph slope down?+
No. Total travel accumulates on outward and return legs. It stays constant during rest
Is a horizontal graph always proof of rest?+
Only under the stated route model. A person moving around a circle could stay equally far from a reference point. These exercises assume one straight route with no hidden reversals
Why can I not use final distance divided by final time?+
For a non-origin segment, use changes between its own endpoints. For average speed over a return journey, use total travel instead of final position
Should rest count when finding average speed?+
Yes, when average speed is requested for the whole journey. Divide total distance by all elapsed time, including rest
What does a vertical segment mean?+
It would change position with no elapsed time, implying unbounded speed. It is not a valid rest or ordinary journey segment
How can I construct a graph from speeds?+
For each interval multiply speed by duration, add for outward motion and subtract for a return, then carry that endpoint into the next interval. Keep units consistent
Is the graph a picture of the road?+
No. Its horizontal coordinate is time, not east-west position. Slope describes the rate of change of the vertical quantity
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 6: Linear graphs