A boat sets off perpendicular to the bank, to cross a river width of 250m. The speed of the boat in still water is 3.20ms-1, however there is a current downstream at 1.20ms-1, which puss the boat sideways.
a) Find the resultant velocity of the boat.
b) Find the total distance the boat travels in crossing the river.
c) How long does the crossing take?
d) How far downstream from the starting point does the boat travel?
a. add the two velocities as vectors.
find the angle of travel down stream
b. distance downstream: you have the angle, and the distance across
c. time=250m/3.20m/s
d. distance downstream=timeinwater/1.20
thanks. the answers to b. is 167m and the answer to d. is 93.7 m. but I'm not sure how to get that. would you please care to explain how to get that?
b. angle: arctan 1.2/3.2
then distance downstreem=250/cosangle
I get for the angle...0.359radians
then for distance = 250/cos.259=259m
Your answer is not possible.
Now another way: time to get across=250/3.2=78.1sec
distance downstream=78.1*1.2=93.7m
now distance total=sqrt(93.7^2+250^2) which is greater than your answer also.
your answer to b is wrong.
a. Vb + Vc = 3.2i + 1.2 = 3.42m/s[69.4o]
= 3.42m/s[20.6o] E. of N.
b. d = 250/Cos20.6 = 267 m.
c. d = V*t = 267 m
3.42*t = 267
t = 78.1 s.
d. d = 250*Tan20.6 = 94 m
To find the solution, we can break down the problem into different components.
a) Finding the resultant velocity of the boat:
To find the resultant velocity, we need to consider the velocity of the boat in still water and the velocity of the current. We can use the Pythagorean theorem to find the magnitude of the resultant velocity.
The magnitude of the resultant velocity can be found using the formula:
Resultant velocity = √(velocity of the boat^2 + velocity of the current^2)
Substituting the given values, we have:
Resultant velocity = √(3.20^2 + 1.20^2)
Resultant velocity = √(10.24 + 1.44)
Resultant velocity = √11.68
Resultant velocity ≈ 3.42 m/s
So, the resultant velocity of the boat is approximately 3.42 m/s.
b) Finding the total distance the boat travels in crossing the river:
To find the total distance, we need to consider the width of the river and the angle between the boat's path and the perpendicular path.
As the boat is traveling perpendicular to the bank, it will form a right-angled triangle with the width of the river as the base and the boat's path as the hypotenuse. The distance traveled by the boat across the river can be found using the formula:
Distance traveled = width of the river / cosine(angle)
The angle can be found using the formula:
Angle = arctan(velocity of the current / velocity of the boat)
Substituting the given values, we have:
Angle = arctan(1.20 / 3.20)
Angle ≈ 0.3589 radians
Distance traveled = 250 / cos(0.3589)
Distance traveled ≈ 250 / 0.933
Distance traveled ≈ 267.98 meters
So, the total distance the boat travels in crossing the river is approximately 267.98 meters.
c) Finding the time taken for the crossing:
To find the time taken, we can use the formula:
Time = Distance / Resultant velocity
Substituting the given values, we have:
Time = 267.98 / 3.42
Time ≈ 78.39 seconds
So, the crossing takes approximately 78.39 seconds.
d) Finding the distance downstream from the starting point:
To find the distance downstream, we can use the formula:
Distance downstream = velocity of the current * time
Substituting the given values, we have:
Distance downstream = 1.20 * 78.39
Distance downstream ≈ 94.07 meters
So, the boat travels approximately 94.07 meters downstream from the starting point.