Sheena can row a boat at 3.13 mi/h in still water. She needs to cross a river that is 1.10 mi wide with a current flowing at 1.75 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?
(d) In order to go straight across, what angle upstream should she have headed?
this sux!!!!!!
Let u : her speed across the stream
Let v : her speed up the stream
Find w : her speed relative to the bank
u = 3.13 cos(60°) [mi/h]
v = 3.13 sin(60°) - 1.35 [mi/h]
Her speed relative to the bank.
w = √(u^2+v^2)
w = √((3.13^2+(3.13√3-1.35)^2)/2) [mi/h]
Time to cross the stream:
Let x : the distance across the bank
Find t : the time to cross the stream
x = 1.10[mi]
t = x/u
t = 1.10/(3.13 cos(60°)) [hr]
Distance traveled up
Find y : the distance upstream
y = vt
y = 1.10(3.13 sin(60°)-1.35)/(3.13 cos(60°)) [mi]
Let θ : angle needed to travel straight across.
Ergo: to have a zero upstream velocity.
0 = 3.13 sin(θ) - 1.35
θ = arcsin(1.35/3.13)
poopy
To solve this problem, we can break it down into several steps:
Step 1: Determine Sheena's speed with respect to the starting point on the bank.
(a) Sheena's speed in still water is given as 3.13 mi/h.
(b) The current speed is given as 1.75 mi/h downstream.
(c) To determine her speed with respect to the starting point, we can use the concept of vector addition. We need to subtract the current speed from her speed in still water in order to account for the effect of the current.
(d) Since Sheena is rowing upstream at a 60.0° angle to the current, the appropriate vector would be the component of her speed perpendicular to the current.
(e) The perpendicular component of Sheena's speed can be calculated using the formula: perpendicular speed = still water speed * sin(θ), where θ is the angle between her desired direction and the current.
(f) Therefore, her speed with respect to the starting point can be calculated as: perpendicular speed = 3.13 mi/h * sin(60.0°).
Step 2: Calculate the time it takes for Sheena to cross the river.
(a) The width of the river is given as 1.10 mi.
(b) To find the time it takes to cross the river, we divide the distance by the speed.
(c) Therefore, the time it takes for Sheena to cross the river can be calculated as: time = distance / speed.
Step 3: Determine how far upstream or downstream from her starting point Sheena will reach the opposite bank.
(a) Since Sheena is rowing upstream at a 60.0° angle, we can calculate the distance she drifts downstream during her crossing using the formula: drift distance = current speed * time taken to cross the river.
(b) If the current speed is positive downstream and negative upstream, and Sheena is rowing at a positive angle upstream, the drift distance should be negative.
(c) Therefore, the distance upstream or downstream from her starting point can be calculated as: drift distance = -current speed * time taken to cross the river.
Step 4: Find the angle Sheena should have headed upstream to go straight across the river.
(a) To find the angle, we can use the formula: θ = sin^(-1)(drift distance / time taken to cross the river).
(b) However, since Sheena already guessed the angle as 60.0° upstream, we can compare her guess with the actual angle we calculated to see if they match.
By following these steps, you can find the answers to all the given questions.