A sail boat is crossing a river of width 43.3 meters, sailing at 17.9 m/s vertically across the river. If a cross wind of 6.0 m/s directed 19.6 degrees off of vertical pushes the sailboat, then how far downstream (horizontally) will the boat be when it reaches the other side of the river? (please provide your answer to 1 decimal place)
A sail boat is crossing a river of width 43.3 meters, sailing at 17.9 m/s vertically across the river. If a cross wind of 6.0 m/s directed 19.6 degrees off of vertical pushes the sailboat, then how far downstream (horizontally) will the boat be when it reaches the other side of the river? (please provide your answer to 1 decimal place)
We can use trigonometry to solve this problem.
First, we can find the velocity of the sailboat in the horizontal direction (downstream) using the cross wind:
V_horizontal = V_wind * cos(angle_off_vertical) = 6.0 m/s * cos(19.6 degrees) = 5.697 m/s
Next, we can use the Pythagorean theorem to find the total speed of the sailboat:
V_total = sqrt((V_sailboat)^2 + (V_horizontal)^2) = sqrt((17.9 m/s)^2 + (5.697 m/s)^2) = 18.771 m/s
Finally, we can use the formula distance = speed * time to find how far downstream the sailboat will be after crossing the river. The time it takes to cross the river can be found using the width of the river and the vertical speed of the sailboat:
time = distance_vertical / V_sailboat = 43.3 m / 17.9 m/s = 2.42 s
Then, the distance downstream can be found using:
distance_downstream = V_horizontal * time = 5.697 m/s * 2.42 s = 13.8 meters
Therefore, the sailboat will be 13.8 meters downstream (horizontally) when it reaches the other side of the river.
To solve this problem, we can use the concept of vector addition.
First, we need to find the horizontal and vertical components of the boat's velocity. The boat's velocity across the river is 17.9 m/s, and the crosswind velocity is 6.0 m/s at an angle of 19.6 degrees off the vertical.
The vertical component of the boat's velocity can be found using trigonometry:
Vertical velocity = 17.9 m/s * sin(19.6 degrees)
Similarly, the horizontal component of the boat's velocity can be found using trigonometry:
Horizontal velocity = 17.9 m/s * cos(19.6 degrees)
Now, to find the time taken by the boat to cross the river, we can use the equation:
Time = Distance / Velocity
The distance the boat needs to cross the river is equal to the width of the river, which is given as 43.3 meters.
Now we can find the distance the boat has traveled downstream when it reaches the other side of the river using the equation:
Distance downstream = Horizontal velocity * Time
Let's calculate the values:
Vertical velocity = 17.9 m/s * sin(19.6 degrees) = 6.08 m/s
Horizontal velocity = 17.9 m/s * cos(19.6 degrees) = 16.95 m/s
Time = 43.3 m / 6.08 m/s = 7.11 s
Distance downstream = 16.95 m/s * 7.11 s = 120.44 meters
Therefore, the boat will be approximately 120.4 meters downstream when it reaches the other side of the river.