a river is 2 km wide and flows at 4km/h. a motorboat has a speed of 10 km/h in still water and heads out from one bank. a marina is directly across the river, on the opposite bank. if the motorboat heads directly toward the marina, how far downstream from the marina will be on the other bank?
Well, it seems like the motorboat is going to be in for a wild ride! But don't worry, I'll try to help you out with some fun calculations.
Let's break down the situation first. The river is 2 km wide and the motorboat has a speed of 10 km/h in still water. However, the river is flowing at a speed of 4 km/h.
So, when the motorboat heads directly toward the marina, it needs to contend with the downstream flow of the river. This means that the effective speed of the boat across the river will be reduced.
To figure out how far downstream the motorboat will be on the other bank, we can use some simple calculations. The ratio between the speed of the boat and the speed of the river will give us an idea of how much the boat will be pushed downstream.
The effective speed of the boat across the river can be calculated as follows:
Effective speed = √(boat speed^2 - river speed^2)
Plugging in the values, we get:
Effective speed = √(10^2 - 4^2)
Effective speed = √(100 - 16)
Effective speed = √84
Effective speed ≈ 9.17 km/h
Now that we know the effective speed of the boat across the river, we can calculate how far downstream it will be on the other bank.
Time = Distance / Speed
Let's assume that the boat reaches the other bank after time 't', and let 'd' be the distance downstream from the marina.
Then we can set up the following equation:
2 km (width of the river) = 9.17 km/h (effective speed) * t (time taken) + d (distance downstream)
Since the boat is heading directly toward the marina, it will reach a point directly across from its starting point. This means that d = 2 km (width of the river).
Substituting this value into the equation, we get:
2 km = 9.17 km/h * t + 2 km
Simplifying, we find:
9.17 km/h * t = 0
Uh-oh! This equation doesn't give us a meaningful solution. It seems like the motorboat won't be able to reach the other bank if it heads directly toward the marina. It's probably going to end up a little downstream from the marina, but not on the other bank I'm afraid!
Well, that was a wild journey! But don't worry, the motorboat will find its way eventually. Maybe it's just taking the scenic route.
To find how far downstream from the marina the motorboat will be on the other bank, we can use the concept of relative velocity.
Let's assume that the motorboat takes time 't' to reach the opposite bank. In this time, the river will also carry the motorboat downstream.
The relative velocity of the motorboat with respect to the water is its actual velocity in still water, which is 10 km/h. Since the river flows at 4 km/h, the relative velocity of the motorboat with respect to the river is 10 km/h - 4 km/h = 6 km/h.
Since the river is 2 km wide, in 't' hours, the motorboat will have traveled a distance of 6 km/h × t = 6t km downstream.
Since the motorboat is heading directly toward the marina, the distance across the river will remain constant. So, the distance traveled by the motorboat perpendicular to the river is the width of the river, which is 2 km.
Using Pythagoras' theorem, we can find the hypotenuse, which represents the distance traveled downstream by the motorboat.
Hypotenuse = √[(Distance downstream)² + (Distance across)²]
Distance downstream = hypotenuse - distance across
Substituting the values, Distance downstream = √[(6t)² + 2²] - 2
Simplifying this equation gives us the distance downstream from the marina on the other bank as √[(36t²) + 4] - 2 km.
To determine how far downstream from the marina the boat will be on the other bank, we need to consider the velocity of the river and the speed of the boat.
First, let's break down the velocities involved:
1. Velocity of the river: The river flows at a speed of 4 km/h.
2. Velocity of the boat in still water: The boat has a speed of 10 km/h in still water.
Now, let's analyze the boat's motion:
When the boat is heading directly toward the marina, it will also be affected by the river's velocity. This is because the boat will be moving diagonally across the river, with the river's flow acting perpendicular to its path.
To find the resulting velocity of the boat, we can use vector addition. The horizontal component of the boat's velocity will be its speed in still water, which is 10 km/h. The vertical component of the boat's velocity will be the river's velocity, which is 4 km/h.
Now we have a right-angled triangle, where the hypotenuse represents the boat's velocity and the two legs represent the horizontal and vertical components.
Using the Pythagorean theorem, we can find the boat's velocity:
Velocity of the boat = √(10^2 + 4^2)
= √(100 + 16)
= √116
= 10.77 km/h (approx.)
Therefore, the boat's velocity, when heading directly toward the marina, is approximately 10.77 km/h.
To calculate the distance downstream from the marina on the opposite bank, we need to consider the time it takes for the boat to cross the river.
Distance downstream = (Velocity of the boat in still water - Velocity of the river) × Time
The time required to cross the river can be determined using the formula:
Time = Distance / Velocity
In this case, the distance is equal to the width of the river, which is 2 km.
Time = 2 km / 10.77 km/h
= 0.1851 hours (approx.)
Now that we have the time, we can calculate the distance downstream:
Distance downstream = (10 km/h - 4 km/h) × 0.1851 hours
= 6 km/h × 0.1851 hours
= 1.11 km (approx.)
Therefore, the motorboat will be approximately 1.11 km downstream from the marina on the opposite bank.
You don't really even need vectors for this. It takes 2/10 hours to cross the river. In that time, the boat drifts downstream 2/10*4 = 4/5 km.
Now, if you want to find the heading needed to get straight across the river, then you need some vectors.