A car is traveling north along a straight road at 50 km/h. An instrument in the car indicates that the wind is coming from the east. If the car's speed is 80 km/h, the instrument indicates that the wind is coming from the northeast. Determine the speed and the direction of the wind.
Well, it seems like this car is in quite the windy situation! Let's see if we can figure out what's going on.
Firstly, we know the car's speed when there is no wind is 50 km/h. But when the car's speed increases to 80 km/h, the wind is coming from the northeast. So, it seems like the wind is somehow helping the car go faster!
To find out the speed of the wind, we can subtract the car's original speed (50 km/h) from its new speed (80 km/h). Math time! 80 km/h minus 50 km/h equals 30 km/h. So, the speed of the wind is 30 km/h.
To determine the direction of the wind, we can picture it as a right-angled triangle, with the car's original and new speeds as the two sides. The wind's direction would be the angle between these two sides.
Since the car's original direction is directly north, and the wind is coming from the east, we can say that the wind is blowing from the northeast. So, my hilarious conclusion is that the wind is coming from the northeast, at a speed of 30 km/h.
Remember, though, this is all just a silly interpretation. The real answer may differ depending on other factors!
To determine the speed and direction of the wind, we can set up a system of equations based on the information given.
Let's assume that the car is traveling along the positive x-axis (east) and the wind is blowing from the positive y-axis (north). We also assume that the car's velocity is relative to the ground.
Let Vc be the velocity vector of the car and Vw be the velocity vector of the wind.
According to the first scenario where the car is traveling north at 50 km/h, we can write:
Vc = (0, 50)
Since the wind is blowing from the east, we can represent it as:
Vw = (vwx, 0)
Now, according to the second scenario where the car is traveling northeast at 80 km/h, we can write:
Vc = (80cos(45), 80sin(45)) = (40√2, 40√2)
In this case, the wind is coming from the northeast, so we can represent it as:
Vw = (vwx, vwy) where vwx > 0 and vwy < 0
We can set up the following equations based on the two scenarios:
(0, 50) + (vwx, 0) = (40√2, 40√2)
This equation represents the vector addition of the car's velocity and the wind's velocity in the second scenario.
Simplifying this equation, we get:
(vwx, 50) = (40√2, 40√2)
Solving for vwx, we get:
vwx = 40√2
Therefore, the x-component of the wind's velocity is 40√2 km/h.
Now, let's calculate the y-component of the wind's velocity using the first scenario equation:
(0, 50) + (40√2, vwy) = (80, 80)
Simplifying this equation, we get:
(40√2, 50 + vwy) = (80, 80)
By equating the x-components and y-components, we can solve for vwy:
40√2 = 80
50 + vwy = 80
Solving these equations, we find:
vwy = -10
Therefore, the y-component of the wind's velocity is -10 km/h.
So, the speed of the wind is √((vwx)^2 + (vwy)^2) km/h, which is:
√((40√2)^2 + (-10)^2) = √(3200 + 100) = √3300 ≈ 57.45 km/h
Finally, we can calculate the direction of the wind using trigonometry. The angle θ between the wind's velocity vector (vwx, vwy) and the positive x-axis is given by:
θ = arctan(vwy / vwx) = arctan(-10 / 40√2) ≈ -14.48°
Hence, the wind is blowing from the northeast at a speed of approximately 57.45 km/h.