A rowboat crosses a river with a velocity
of 1.01 m/s at an angle 30◦ North of West
relative to the water.
Water’s frame of reference:
speed of boat: 1.01 m/s
direction from current: 30◦
The river is 322mwide and carries a current
of 1.63 m/s due East.
Shore’s frame of reference:
current of water:1.63 m/s
distance shore to shore: 322 m
When the boat reaches the opposite bank,
how far East or West (downstream “+” or
upstream “−”) from its starting point (on the
opposite bank) does it land.
Note: In your answer, take an Eastward
displacement to be positive and a Westward
displacement to be negative.
Answer in units of m.
Well, I must say, that's a rather challenging question! Let me put on my thinking cap and calculate the answer with a touch of humor.
First, let's figure out the boat's velocity relative to the shore. Since the boat is moving north of west at 30 degrees, we can break down its velocity into its east and north components.
The eastward component of the boat's velocity is given by 1.01 m/s * cos(30 degrees). And the northward component is given by 1.01 m/s * sin(30 degrees).
So, the eastward velocity relative to the shore is 1.01 m/s * cos(30 degrees), which equals approximately 0.876 m/s. And the northward velocity relative to the shore is 1.01 m/s * sin(30 degrees), which equals approximately 0.505 m/s.
Now we need to consider the river's current, which is flowing east at 1.63 m/s. This means we need to subtract 1.63 m/s from the boat's eastward velocity.
So, the boat's resulting eastward velocity relative to the shore is approximately 0.876 m/s minus 1.63 m/s. Now we're getting there!
To find out how far the boat lands east or west of its starting point, we need to determine the time it takes to cross the river. The distance between both riverbanks is given as 322 m.
The time it takes to cross the river is equal to the distance divided by the boat's eastward velocity. So, we have 322 m divided by (0.876 m/s - 1.63 m/s).
Now, I'll do some quick calculation... Crunch, crunch, crunch... and voila! The time comes out to be approximately 632.21 seconds.
Finally, to determine the boat's east or west displacement, we multiply the boat's northward velocity (0.505 m/s) by the time (632.21 seconds).
Allow me to calculate that for you... Carry the one, add a pinch of salt... and tada! The answer is approximately -319.99 meters.
So, the boat lands about 320 meters west of its starting point on the opposite bank. Remember, taking a westward displacement as negative means it's pulling back a bit. Who knew rivers could be so crafty?
To determine how far east or west the boat lands, we need to find the horizontal displacement caused by both the velocity of the boat and the current of the river.
First, let's find the boat's horizontal displacement. We can use the velocity magnitude (1.01 m/s) and the angle (30 degrees north of west) to find the boat's horizontal velocity. We can use trigonometry to find that the horizontal component of the velocity is given by:
Vboat_horizontal = velocity * cos(angle)
Vboat_horizontal = 1.01 m/s * cos(30 degrees)
Vboat_horizontal = 1.01 m/s * √(3)/2
Vboat_horizontal = 0.876 m/s
Next, let's find the river's horizontal displacement. The current of the river is given as 1.63 m/s due east. The time it takes for the boat to cross the river is the same amount of time it takes the boat to move downstream. Therefore, the horizontal displacement caused by the river's current is:
Drift_horizontal = current * time
Drift_horizontal = 1.63 m/s * time
Now, we need to find the time it takes for the boat to cross the river. To do this, we can use the width of the river (322 meters) and the boat's horizontal velocity:
time = distance / velocity
time = 322 m / 0.876 m/s
time = 368 seconds
Now, we can determine the horizontal displacement caused by the river:
Drift_horizontal = 1.63 m/s * 368 s
Drift_horizontal = 600.24 m
To find the total horizontal displacement, we need to add the boat's displacement and the river's displacement:
Total_horizontal_displacement = Vboat_horizontal + Drift_horizontal
Total_horizontal_displacement = 0.876 m/s + 600.24 m
Total_horizontal_displacement = 601.116 m
Since the boat is moving west relative to the shore's frame of reference and we are considering westward displacement as negative, the boat lands approximately 601.116 meters west (or -601.116 m) from its starting point on the opposite bank.