A boat travels at 3.8 m/s and heads straight across a river 240m wide at an angle of 145'. The river flows at 1.6 m/s.
a. What is the boat's resultant speed with respect to the river bank?
b. How long does ti take the boat to cross the river?
c. How far downstream or upstream is the bot when it reaches the other side?
To solve this problem, we can break down the boat's motion into two components: its speed perpendicular to the river's flow (across the river) and its speed parallel to the river's flow (upstream or downstream).
a. To find the boat's resultant speed with respect to the river bank, we can use the Pythagorean theorem. The boat's velocity across the river is given by its speed multiplied by the cosine of the angle between its path and the river bank. So, the velocity across the river is:
velocity across the river = speed × cos(angle)
Substituting the given values, we have:
velocity across the river = (3.8 m/s) × cos(145°)
Now, the velocity downstream or upstream is given by the river's flow velocity minus the boat's velocity across the river. So, the velocity downstream or upstream is:
velocity downstream or upstream = river flow velocity - velocity across the river
Substituting the given values, we have:
velocity downstream or upstream = (1.6 m/s) - (3.8 m/s × cos(145°))
b. To find how long it takes for the boat to cross the river, we can use the formula:
time = distance / velocity across the river
Substituting the given values, we have:
time = 240m / (3.8 m/s × cos(145°))
c. To find how far downstream or upstream the boat is when it reaches the other side, we can use the formula:
distance = velocity downstream or upstream × time
Substituting the values we found in part a for the velocity downstream or upstream, and the value we found in part b for the time, we have:
distance = [(1.6 m/s) - (3.8 m/s × cos(145°))] × (240m / (3.8 m/s × cos(145°)))