1. If the angle of the boat is 150 degrees, v' = 3 m/s, and v0 = 3.0 m/s, how far do you drift downstream before you arrive at the other side of the river? (HINT: read the x axis.)
2. If the river flow rate (v0) = 3.0 m/s, and the angle (è, or “theta”) is 150 degrees, what v' (velocity of the boat) would be required to end up directly across the river?
3. If the angle remains 150 degrees, and the boat velocity (v') is 3.0 m/s, but the river flow rate is increased to 4.0 m/s, how far would you drift downstream before you arrived at the other side of the river?
1. To find out how far you drift downstream before arriving at the other side of the river, we can use trigonometry and the given information.
First, let's break down the problem. We have the angle of the boat (150 degrees) and two velocities: v' (boat velocity) and v0 (river flow rate velocity). We need to find the distance drifted downstream.
To solve this, we will use the concept of vector addition. The boat velocity and the river flow rate velocity can be considered as two vectors. The resultant vector will be the diagonal of the parallelogram formed by these two vectors.
Since the angle is 150 degrees, we can use the trigonometric function cosine to find the horizontal component (x-axis) of the resultant vector.
The formula to find the horizontal component is:
horizontal component = v' * cos(angle)
Substituting the given values:
horizontal component = 3 m/s * cos(150 degrees)
To calculate the horizontal component, we need to convert the angle from degrees to radians since trigonometric functions typically take radians as input.
Using the conversion, 150 degrees = (150 * pi) / 180 radians.
So, horizontal component = 3 m/s * cos((150 * pi) / 180 radians)
Calculating this value will give us the horizontal component.
Once we have the horizontal component, we can use the formula of distance (d) traveled in a given time (t) as:
distance = horizontal component * time
In this case, time can be considered as the time it takes to cross the river. So, distance gives us the drift downstream before arriving at the other side of the river.
2. To determine the velocity of the boat (v') required to end up directly across the river, we need to consider the angle and the river flow rate. We want the resultant vector, considering both the boat's velocity and the river flow rate, to have a horizontal component equal to 0 (moving directly across the river).
Using the same formula as in question 1, we have:
horizontal component = v' * cos(angle) + v0 * cos(90 - angle)
Since we want the horizontal component to be zero, we can set the entire equation to zero and solve for v'. This will give us the required velocity of the boat to end up directly across the river.
3. If the angle remains 150 degrees, the boat velocity (v') is 3.0 m/s, and the river flow rate is increased to 4.0 m/s, we can follow the same steps as in question 1 to find the distance drifted downstream.
Using the increased river flow rate (v0 = 4.0 m/s), we can calculate the new horizontal component of the resultant vector. We then use the same formula to calculate the distance drifted downstream by multiplying the horizontal component by the time it takes to cross the river.