A boat travels at a speed of 5 m/s in still water. The boat moves directly across a river that is 70m wide. The water in the river flows at a speed of 2 m/s. How long does it take the boat to cross the river? In what direction is the boat headed when it starts the crossing.
Textbook Answer: 15.3 m/s, 66.4 degrees to shore
My answer: 14 seconds and 22 degrees from the shore.
This is what I did: I found the speed of the resultant by doing the Pythagorean (5.4 m/s),then found the distance of the resultant by "5/70 = 2/x (28m)" and then using the Pythagorean (75.4m), and then I found the time by doing "t = d/s". Then, I found the direction of the boat by using trig, "tanx = 2/5"
-------------MY QUESTION: What did I do wrong? And can you please help me fix my mistake?
How can the textbook answer be 15.3 m/s if they are asking for the crossing time?
To go directly across, the boat must aim upstream so that the velocity component upstream relative to the water is 2 m/s. That makes its velocity component across the water sqrt[5^2 - 2^2) = sqrt 21 = 4.58 m/s. The crossing time is
70 m/4.58 m/s = 15.3 s. The pointing angle of the boat is cos^-1 4.58/5 = 23.7 degrees relative to the cross-stream direction, or 66.3 degrees relative to the shore.
If the boat moves directly across then there is enough of the boats speed vector to counteract the river flow. That is 2m/s.
So, the vector parallel to the shore (the river flow) speed is 2m/s. The TOTAL speed of the boat is 5m/s. Now solve this for the speed perpendicular to the shore.
SQRT(5^2 - 2^2).
Divide that into the distance (70m).
Use any of the trig values to find the angle.
Your calculation of the time taken to cross the river is incorrect. Let me explain the correct steps:
To determine the time taken to cross the river:
1. Calculate the speed of the resultant velocity of the boat. This can be found using the Pythagorean theorem:
Resultant velocity = √(boat velocity^2 + river velocity^2)
= √(5^2 + 2^2)
= √29
≈ 5.39 m/s
2. Calculate the distance it needs to travel across the river:
Distance = river width
= 70 m
3. Now, you can calculate the time taken using the formula:
Time = Distance / Speed
= 70 m / 5.39 m/s
≈ 13 seconds (rounded to the nearest whole number)
Therefore, the correct answer is approximately 13 seconds, not 14 seconds as you calculated.
Regarding the direction of the boat when it starts the crossing, the textbook answer is 66.4 degrees to the shore. This means the boat is not directly perpendicular to the shore but at an angle of 66.4 degrees.
Your approach is generally correct, but there seems to be a mistake in one of your calculations.
To solve the problem correctly, you need to consider the vector addition of the boat's velocity in still water and the velocity of the river flow. This will give you the resulting velocity of the boat relative to the shore.
First, let's find the perpendicular component of the boat's velocity to cross the river. To cross the river of width 70m, the boat needs to move perpendicular to the shore. Using the given information, the boat's perpendicular velocity can be determined using the Pythagorean theorem:
Perpendicular velocity = √(5^2 - 2^2) = √21 m/s
Next, let's find the time it takes for the boat to cross the river. This can be done by dividing the width of the river by the perpendicular velocity:
Time = Distance / Perpendicular velocity = 70m / (√21 m/s) ≈ 9.4 seconds
So it takes approximately 9.4 seconds for the boat to cross the river.
Regarding the direction of the boat, you can find the angle by using trigonometry. The angle can be calculated as the inverse tangent of the ratio of the perpendicular velocity to the parallel velocity (in the direction of the river flow):
Angle = arctan(2 m/s / 5 m/s) ≈ 21.8 degrees
Therefore, the boat is headed at an angle of approximately 21.8 degrees from the direction opposite to the river flow.
Based on the textbook answer provided, it seems there may be a discrepancy with the values. The resultant speed of 15.3 m/s and 66.4 degrees might not be correct for this particular problem.