Two canoeists in identical canoes exert the same effort paddling and hence maintain the same speed relative to the water. One paddles directly upstream (and moves upstream), whereas the other paddles directly downstream. With downstream as the positive direction, an observer on shore determines the velocities of the two canoes to be
−1.7 m/s
and +2.5 m/s, respectively.
(a) What is the speed of the water relative to the shore?
m/s
(b) What is the speed of each canoe relative to the water?
m/s
The driver of a speeding empty truck (initial speed of Vi) slams on the brakes and skids to a stop through a distance d. Assuming that the brakes always supply the same stopping force, consider the following situations:
1
(a) If the truck carried a load that doubled its mass, what would be the
truck’s “skidding distance” in terms of d?
2d
d
d/2
d/4
2
(b) If the initial speed (with its initial mass) of the truck were halved, what would be the truck’s “skidding distance” in terms of d?
2d
d
d/2
d/4
3
(c) If the stopping distance is now 4d, what is the initial velocity (Vi) of the truck?
2Vi
Vi
Vi/2
Vi/4
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To answer these questions, we can apply the concept of relative velocity. The speed of the water relative to the shore can be determined by considering the velocities of the canoes as observed from the shore.
Let's denote the velocity of the water as Vw and the velocity of each canoe relative to the water as Vc.
(a) To find the speed of the water relative to the shore, we can use the fact that the velocity of each canoe relative to the shore is the vector sum of its velocity relative to the water and the velocity of the water relative to the shore.
For the canoe paddling upstream:
Velocity of canoe relative to shore = Velocity of canoe relative to water - Velocity of water relative to shore
Vcs = Vcu - Vw
Substituting the given values:
Vcs = -1.7 m/s - Vw
For the canoe paddling downstream:
Velocity of canoe relative to shore = Velocity of canoe relative to water + Velocity of water relative to shore
Vcs = Vcd + Vw
Substituting the given values:
Vcs = 2.5 m/s + Vw
Since both canoes exert the same effort and maintain the same speed relative to the water, we can set their velocities relative to the shore equal to each other:
-1.7 m/s - Vw = 2.5 m/s + Vw
Solving this equation, we find:
-1.7 m/s - 2.5 m/s = 2Vw
-4.2 m/s = 2Vw
Vw = -2.1 m/s
Therefore, the speed of the water relative to the shore is 2.1 m/s.
(b) To find the speed of each canoe relative to the water, we can use the equation we derived in part (a) and substitute the value of Vw we just found:
For the canoe paddling upstream:
Vcu = Vcs + Vw
Vcu = -1.7 m/s + (-2.1 m/s)
Vcu = -3.8 m/s
For the canoe paddling downstream:
Vcd = Vcs - Vw
Vcd = 2.5 m/s - (-2.1 m/s)
Vcd = 4.6 m/s
Therefore, the speed of each canoe relative to the water is:
Vcu = -3.8 m/s (upstream)
Vcd = 4.6 m/s (downstream)