SME
Grade 11 Science · Problem type
A boat crossing a river
Use this for “A boat heads straight across a 120 m river at 4 m/s; the current is 3 m/s. How long does it take, and how far downstream does it land?”
Diagram
boat: 4 m/s at 90°, so boat = ⟨0, 4⟩.
current: 3 m/s at 0°, so current = ⟨3, 0⟩.
Tip to tail: v = ⟨3, 4⟩
|v| = 5 m/s at 53.13°.
River width: 120 m
Crossing time: 30 s
Drift downstream: 90 m
This is an example. Type your own values to start a new problem.
b
Boat’s speed in still water
example
c
Current’s speed
example
v
Speed over the ground
calculated
θ
Direction, from east
calculated · °
W
River width
example
t
Crossing time
calculated
s
Drift downstream
calculated
Formulas
v
=
√
b
2
+
c
2
5
=
√
4
2
+
3
2
θ
=
arctan(
b
c
)
53.1
=
arctan(
4
3
)
t
=
W
b
30
=
120
4
s
=
c
×
t
90 = 3 × 30
Assumptions
•
The boat heads straight across (north); the current flows east.
•
Its velocity over the ground is the vector sum of the two: they are at right angles.
•
The current doesn’t slow the crossing: the time is the width over the boat’s own speed.
Step-by-step
WE KNOW
b = 4 m/s, c = 3 m/s, W = 120 m
FIND
speed over the ground (v), direction, from east (θ), crossing time (t), drift downstream (s)
Step 1 · Find speed over the ground (v)
Use v = √(b² + c²)
The two velocities are at right angles: add them by Pythagoras.
v
=
√
4
2
+
3
2
v
=
√
16 + 9
v
=
√
25
v
=
5
m/
s
Step 2 · Find direction, from east (θ)
Use θ = arctan(b/c)
The tangent of the angle from east is north over east.
θ
=
arctan(
4
3
)
θ
=
53.1°
Step 3 · Find crossing time (t)
Use t = W/b
Only
the
boat’
s
own
speed
carries
it
across:
the
width
over
that
speed.
t
=
120
4
t
=
30
s
Step 4 · Find drift downstream (s)
Use s = c × t
All
the
while,
the
current
carries
it
east:
the
current’
s
speed
times
the
time.
s
=
3
×
30
s
=
90
m
Check
Put the numbers back in. Both sides should match.
5
=
√
4
2
+
3
2
✓
53.1
=
arctan(
4
3
)
✓
30
=
120
4
✓
90 = 3 × 30 ✓
Topic check
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