O-LevelAdditional MathematicsVectors in two dimensions4037/22 · Oct/Nov 2013 · Q98 Marks
The diagram shows a river with parallel banks. The river is 40m wide and is flowing with a speed of 1.8 ms⁻¹. A canoe travels in a straight line from a point P on one bank to a point Q on the opposite bank 70m downstream from P. Given that the canoe takes 12s to travel from P to Q, calculate the speed of the canoe in still water and the angle to the bank that the canoe was steered. [Figure showing river with width 40m, downstream distance 70m, river flow 1.8 m s⁻¹, path from P to Q]
- (a)The diagram shows a river with parallel banks. The river is 40m wide and is flowing with a speed of 1.8 ms⁻¹. A canoe travels in a straight line from a point P on one bank to a point Q on the opposite bank 70m downstream from P. Given that the canoe takes 12s to travel from P to Q, calculate the speed of the canoe in still water and the angle to the bank that the canoe was steered.[8]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.
- [M1] Takes components
- [A1] 12v sin α = 40
- [A1] 12(v cos α + 1.8) = 70
- [M1A1] 12v cos α = 48.4
- [DM1] Solve for v or α
- [A1] α = 39.6 Allow 0.691 radians
- [A1] v = 5.23
- Method B
- [B1] x=1.8×12= 21.6
- [B1] y = 70 - 21.6 = 48.4
- [M1] D² = 40² + 48.4² (= 3942.56)
- [A1] D = 62.8
- [DM1] V = D/12
- [A1] V = 5.23 5.23 or better
- [M1] tan α = 40/48.4
- [A1] α = 39.6° Allow 0.691 radians
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