Sheena can row a boat at 2.90 mi/h in still water. She needs to cross a river that is 1.22 mi wide with a current flowing at 1.55 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?
To solve this problem, we can use the concept of vector addition and trigonometry.
(a) To find Sheena's speed with respect to the starting point on the bank, we need to calculate the resultant velocity. The resultant velocity is the vector sum of the boat's velocity in still water and the velocity of the current. We can use the Pythagorean theorem to find the magnitude of the resultant velocity.
Let's denote the velocity of the boat in still water as v_boat (2.90 mi/h) and the velocity of the current as v_current (1.55 mi/h).
The magnitude of the resultant velocity, v_resultant, can be calculated as:
v_resultant = sqrt((v_boat)^2 + (v_current)^2)
Plugging in the given values:
v_resultant = sqrt((2.90 mi/h)^2 + (1.55 mi/h)^2)
= sqrt(8.41 mi^2/h^2 + 2.40 mi^2/h^2)
= sqrt(10.81 mi^2/h^2)
= 3.29 mi/h
Therefore, Sheena's speed with respect to the starting point on the bank is approximately 3.29 mi/h.
(b) To find the time it takes for Sheena to cross the river, we can use the formula:
time = distance / velocity
The distance Sheena needs to cross is the width of the river, which is 1.22 mi. And we already know her speed with respect to the starting point on the bank, which is 3.29 mi/h.
Plugging in the values:
time = 1.22 mi / 3.29 mi/h
= 0.37 h
Therefore, it takes Sheena approximately 0.37 hours to cross the river.
(c) To find how far upstream or downstream from her starting point Sheena will reach the opposite bank, we can use trigonometry.
Let's denote the distance we are interested in as d. We can use the relation:
d = velocity of the current * time
Plugging in the values:
d = 1.55 mi/h * 0.37 h
= 0.57 mi
Therefore, Sheena will reach the opposite bank approximately 0.57 miles downstream from her starting point.
(d) In order to go straight across, Sheena should have headed directly opposite to the direction of the current. Since she guessed 60.0° upstream, we need to find the angle that is opposite to 60.0°.
The angle upstream, theta, can be calculated as:
theta = 180° - 60.0°
= 120°
Therefore, in order to go straight across the river, Sheena should have headed 120° upstream.