Trevor can row a boat at 3.00 mi/hr in still water. He needs to cross a river that is 1.20 miles wide with a current flowing at 1.60 mi/h. Not having his calculator ready, he guesses to go straight across, he should head 60.0 degrees upstream with respect to straight across. a) what is his speed with respect to the starting point on the shore? b) how long does it take to cross the river? c) what should the angle have been to have gone directly across the river?
To solve this problem, we can break it down into three parts:
a) To find Trevor's speed with respect to the starting point, we can use the concept of vector addition. Trevor's speed with respect to the ground can be calculated by finding the resultant vector of his rowing speed and the river current:
Speed with respect to the ground = square root((speed in still water)^2 + (speed of current)^2 + 2 * (speed in still water) * (speed of current) * cos(angle between them))
Given:
- Speed in still water (Trevor's rowing speed) = 3.00 mi/hr
- Speed of current = 1.60 mi/hr
- Angle between the rowing direction and current direction = 60.0 degrees
Plugging the values into the formula, we get:
Speed with respect to the ground = square root((3.00)^2 + (1.60)^2 + 2 * 3.00 * 1.60 * cos(60.0))
Now, let's calculate the value:
Speed with respect to the ground = square root(9 + 2.56 + 9.6 * cos(60.0))
Speed with respect to the ground = square root(11.56 + 9.6 * 0.5)
Speed with respect to the ground = square root(11.56 + 4.8)
Speed with respect to the ground = square root(16.36)
Speed with respect to the ground ≈ 4.05 mi/hr
Therefore, Trevor's speed with respect to the starting point on the shore is approximately 4.05 mi/hr.
b) To find the time it takes to cross the river, we can use the formula:
Time = Distance / Speed
Given:
- Width of the river = 1.20 miles
- Speed with respect to the ground = 4.05 mi/hr
Plugging the values into the formula, we get:
Time = 1.20 / 4.05
Now, let's calculate the value:
Time ≈ 0.30 hours
Therefore, it takes approximately 0.30 hours (or 18 minutes) to cross the river.
c) To find the angle Trevor should have aimed to go directly across the river, we can use the concept of vectors again. The angle between the resultant vector (speed with respect to the ground) and the current direction is the angle Trevor should have aimed for.
Given:
- Angle between the rowing direction and current direction = 60.0 degrees
To find the angle Trevor should have aimed for, we need to find the supplement angle to the result of part c). The angle is given by:
Angle = 180 - 60.0
Now, let's calculate the value:
Angle = 180 - 60.0
Angle = 120.0 degrees
Therefore, Trevor should have aimed at an angle of 120.0 degrees to go directly across the river.