Sheena can row a boat at 2.91 mi/h in still water. She needs to cross a river that is 1.24 mi wide with a current flowing at 1.54 mi/h. Not having her calculator ready, she guesses that to go straight across, she should head 60.0° upstream.
(a) What is her speed with respect to the starting point on the bank?
(b) How long does it take her to cross the river?
(c) How far upstream or downstream from her starting point will she reach the opposite bank?
magnitude
direction
(d) In order to go straight across, what angle upstream should she have headed????
To solve this problem, we need to break it down into different parts and use a few concepts in physics, such as vector addition and trigonometry.
(a) Let's first find Sheena's speed with respect to the starting point on the bank. We can do this by considering the velocity vectors involved. Sheena's boat speed, 2.91 mi/h, represents her velocity in still water. The current speed, 1.54 mi/h, represents the velocity of the river current.
Since Sheena is rowing upstream, her velocity relative to the river current will be the vector difference between her boat's velocity and the current velocity. Therefore, her speed with respect to the starting point on the bank is calculated as follows:
Speed with respect to starting point = (Boat speed)^2 - (Current speed)^2
= (2.91 mi/h)^2 - (1.54 mi/h)^2
Calculate the square of each speed: (2.91)^2 = 8.4681 mi^2/h^2 and (1.54)^2 = 2.3716 mi^2/h^2.
Now subtract the current velocity component from the boat velocity component:
Speed with respect to starting point = (8.4681 mi^2/h^2 - 2.3716 mi^2/h^2)^0.5
= (6.0965 mi^2/h^2)^0.5
≈ 2.47 mi/h
Thus, Sheena's speed with respect to the starting point on the bank is approximately 2.47 mi/h.
(b) To find out how long it takes her to cross the river, we can use the relationship:
Time = Distance / Speed
The distance she needs to cross the river is given as 1.24 mi, and her speed with respect to the starting point on the bank is 2.47 mi/h. Therefore:
Time = 1.24 mi / 2.47 mi/h
≈ 0.502 h
It will take her approximately 0.502 hours, or approximately 30.1 minutes, to cross the river.
(c) Now, let's calculate how far upstream or downstream from her starting point she will reach the opposite bank.
Since she is rowing at an angle of 60.0° upstream, and the river's current pushes her downstream, we need to calculate the resultant displacement.
Horizontal displacement = (Speed with respect to starting point) * (Time)
= (2.47 mi/h) * (0.502 h)
≈ 1.24 mi
Therefore, she will reach the opposite bank at the same horizontal distance of 1.24 mi from her starting point.
To calculate the magnitude of the displacement, we can use the Pythagorean theorem.
Magnitude of displacement = √((Horizontal displacement)^2 + (Distance of the river width)^2)
= √((1.24 mi)^2 + (1.24 mi)^2)
= √(1.5376 mi^2 + 1.5376 mi^2)
= √3.0752 mi^2
≈ 1.75 mi
So, she will reach the opposite bank at approximately a distance of 1.75 miles upstream or downstream from her starting point.
To find the direction, we can use trigonometry. The tangent of the angle formed between the horizontal and the resultant displacement can give us the direction.
Direction = atan((Distance of the river width) / (Horizontal displacement))
= atan(1.24 mi / 1.24 mi)
= atan(1)
≈ 45°
Therefore, she will reach the opposite bank at an angle of approximately 45° upstream or downstream from her starting point.
(d) To go straight across, she should have headed at an angle upstream that is equal to the angle formed between the horizontal and the resultant displacement. As calculated in part (c), the angle is approximately 45°.
Hence, to go straight across, Sheena should have headed at an angle of approximately 45° upstream.