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v
a
v + w
w
20
triangles, Solving this proportion, 0 v ! w ! 0 20.9, so the
2 0v! w ! 0
2.09 .
actual speed of the boat crossing the river was about 20.9 km>h. The actual
time taken to cross the river is t d or about 6 min.
v 2.09
20.9 0.1 h,
Therefore, the boat landed 0.6 km downstream, and it took approximately
6 min to make the crossing.
c. To determine the velocity with which she must travel to reach the marina, we
will draw the related vector diagram.
We are given 0w ! 0 6 and 0v ! 0 20. To determine the direction in which
the boat must travel, let a represent the angle upstream at which the boat
heads out.
sin a 6 20 or sin1 a 6 20 b a
a 17.5°
To calculate the magnitude of the resultant velocity, use the Pythagorean theorem.
0 v ! 0 2 0 w ! 0 2 0 v ! w ! 2
0 where 0v ! 0 20 and 0w ! 0 6
Thus, 20 2 6 2 0 v ! w ! 2
0
0 v ! w ! 0 2 400 36
0 v ! w ! 0 19.08
This implies that if Anna wants to travel directly across the river, she will have
to travel upstream 17.5° with a speed of approximately 19.08 km>h. The nose
of the boat will be headed upstream at 17.5°, but the boat will actually be
moving directly across the river at a water speed of 19.08 km>h.
IN SUMMARY
Key Idea
• Problems involving velocities can be solved using strategies involving vectors.
Need to Know
• The velocity of an object is stated relative to a frame of reference. The frame
of reference used influences the stated velocity of the object.
• Air speed/water speed is the speed of a plane/boat relative to a person on
board. Ground speed is the speed of a plane or boat relative to a person on
the ground and includes the effect of wind or current.
! ! !
• The resultant velocity v r v1 v2 .
368 7.2 VELOCITY
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