A river 400 m wide is flowing at a rate of 2.0 m/s. A boat is sailing at a velocity of 10 m/s with respect to the wate, in a direction perpendicular to the river. (a) Find the time taken by the boat to reach the opposite bank. (b) How far from the point directly opposite to the starting point does the boat reach the opposite bank?

Question

A river 400 m wide is flowing at a rate of 2.0 m/s. A boat is sailing at a velocity of 10 m/s with respect to the wate, in a direction perpendicular to the river. (a) Find the time taken by the boat to reach the opposite bank. (b) How far from the point directly opposite to the starting point does the boat reach the opposite bank?

Answer 1

the downstream problem is perpendicular to and therefore independent of (except in time) the across problem

400 = 10 t
t = 40 seconds

then

d = 2 t = 80 m

Answer 2

To find the time taken by the boat to reach the opposite bank, we can use the concept of relative velocity.

(a) Speed of the boat with respect to the water = 10 m/s
Speed of the river = 2.0 m/s

Since the boat is moving perpendicular to the river, the effective velocity of the boat (resultant velocity) will be the vector sum of the boat's velocity with respect to the water and the river's velocity.

Using the Pythagorean theorem, we can find the effective velocity of the boat:
Effective velocity = √((velocity of boat)^2 + (velocity of river)^2)
= √((10 m/s)^2 + (2 m/s)^2)
= √(100 m^2/s^2 + 4 m^2/s^2)
= √(104 m^2/s^2)
≈ 10.2 m/s

Now, we can find the time taken by the boat to cross the river using the formula:
Time = Distance / Speed

Distance to cross the river = width of the river = 400 m
Speed of the boat with respect to the water = 10 m/s

Time taken by the boat to cross the river = Distance / Speed
= 400 m / 10.2 m/s
≈ 39.2 s

Therefore, the time taken by the boat to reach the opposite bank is approximately 39.2 seconds.

(b) To find how far from the point directly opposite to the starting point the boat reaches the opposite bank, we can use the formula:
Distance = Speed x Time

Speed of the river = 2.0 m/s
Time taken by the boat to cross the river = 39.2 s

Distance from the point directly opposite to the starting point to the opposite bank = Speed of the river x Time
= 2.0 m/s x 39.2 s
≈ 78.4 m

Therefore, the boat reaches a distance of approximately 78.4 meters from the point directly opposite to the starting point when it reaches the opposite bank.