A boat travels at 3.8 m/s and heads straight across a river 240 m wide. The river flows at 1.6 m/s. Find
A) The boat's resultant velocity.
B) How long it takes the boat to cross the river.
C) How far downstream the boat is when it reaches the other side.
Please help, i have about 12 of these problems that I understand NONE of.
The time it takes to get across is 240m divided by 3.8 m/s, which is 63.2 seconds. In that time the current has moved boat downstream by: 1.6 m times 63.2 seconds, or 101.1 meters. Then to get the resultant velocity, you need to add the velocity vectors. Use pythagorean theorem: resultant^2 = 3.8^2 + 1.6^2. The resultant velocity is 4.1 m/s. Does this make sense??
Lazy Laz
No problem, I'm here to help you step-by-step with these problems. Let's start with the given information:
Boat's velocity relative to the ground (v_b) = 3.8 m/s
River's velocity (v_r) = 1.6 m/s
River's width (d) = 240 m
A) To find the boat's resultant velocity, we need to use vector addition. The boat's resultant velocity (v_resultant) can be found using the Pythagorean Theorem.
v_resultant^2 = v_b^2 + v_r^2
v_resultant^2 = (3.8)^2 + (1.6)^2
v_resultant^2 = 14.44 + 2.56
v_resultant^2 = 17
v_resultant = sqrt(17)
v_resultant ≈ 4.12 m/s
B) To find how long it takes the boat to cross the river, we can use the formula distance = speed × time. The boat needs to cover the width of the river (d = 240 m).
time = distance / velocity_relative
time = 240 m / 3.8 m/s
time ≈ 63.16 s
Therefore, it takes approximately 63.16 seconds for the boat to cross the river.
C) To find how far downstream the boat is when it reaches the other side, we can calculate the horizontal displacement caused by the river's flow. The downstream displacement (horizontal distance traveled downstream) is given by:
downstream_displacement = river_velocity × time
downstream_displacement = 1.6 m/s × 63.16 s
downstream_displacement ≈ 101.06 m
So, when the boat reaches the other side of the river, it will be approximately 101.06 meters downstream from its initial position.
I hope this helps! Let me know if you have any further questions.
Sure! I'll be happy to help you understand these problems step by step.
A) To find the boat's resultant velocity, we need to use vector addition since the boat's velocity and the river's velocity are acting in different directions.
The resultant velocity (v) can be found using the Pythagorean theorem: v = √(vboat^2 + vriver^2), where vboat is the magnitude of the boat's velocity and vriver is the magnitude of the river's velocity.
In this case, vboat = 3.8 m/s and vriver = 1.6 m/s.
So, v = √(3.8^2 + 1.6^2) = √(14.44 + 2.56) = √17 = 4.12 m/s (rounded to two decimal places).
Therefore, the boat's resultant velocity is approximately 4.12 m/s.
B) To find how long it takes the boat to cross the river, we can use the formula: time = distance / velocity.
The distance the boat needs to cross is given as 240 m, and the boat's resultant velocity is 4.12 m/s (as calculated above).
So, the time taken to cross the river = 240 m / 4.12 m/s ≈ 58.25 s (rounded to two decimal places).
Therefore, it takes approximately 58.25 seconds for the boat to cross the river.
C) To find how far downstream the boat is when it reaches the other side, we can use the formula: distance = velocity x time.
The velocity we'll use here is the river's velocity, vriver = 1.6 m/s, since the boat would drift downstream due to the river's flow after crossing the river.
We already know the time taken to cross the river is 58.25 s (as calculated above).
So, the distance downstream the boat travels = 1.6 m/s x 58.25 s ≈ 93.20 m (rounded to two decimal places).
Therefore, the boat is approximately 93.20 meters downstream from its starting point when it reaches the other side of the river.
I hope this explanation helps! Let me know if you have any more questions.