- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Differentiation rules practice question 17
A particle moves along a line with position metres at time seconds. Find the total distance it travels between and .
Distance counts every metre, forwards or backwards. Find where the velocity changes sign between t = 0 and t = 3: the particle turns there.
, so the particle turns at . Find , and .
Add the lengths of the legs between the turning points, each taken as a positive number.
A number of metres.
Theory UBC Math 100 §4.1CLP-1 §2.4Key results
Worked solution
Strategy
Total distance differs from displacement when the particle turns round. Find the turning points from v = 0, then add the distance covered on each leg.
Working
Differentiate and factor.
The particle moves forward until t = 2, then back; at t = 3 it stops again.
Positions at the start, the turn and the end.
Add the length of each leg.
Answer
The answer is 29 metres.
Check
29 m exceeds the 27 m displacement, as it must: v(2.5) = −1.5 m/s, so the particle backs up after t = 2.
Where this goes wrong
Answering s(3) − s(0) = 27, the displacement. The particle goes forward 28 m and back 1 m, so it travels 29 m in all.
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