An airplane flies due north at 250 km/h relative to the air. There is a wind blowing at 60 km/h to the northeast relative to the ground. What are the plane's speed and direction relative to the ground?

To find the plane's speed and direction relative to the ground, we can use vector addition.

Let's break down the velocities into their horizontal (east-west) and vertical (north-south) components.

The airplane's velocity relative to the air is due north at 250 km/h. So, its vertical component is 250 km/h (north) and its horizontal component is zero km/h (east-west).

The wind's velocity relative to the ground is blowing at 60 km/h to the northeast. We need to split this velocity into its horizontal and vertical components. To do that, we can use trigonometry.

The direction of the wind can be represented as a 45-degree angle to the positive x-axis (east) since it is blowing towards the northeast. Using basic trigonometry, we can determine that the horizontal and vertical components of the wind's velocity are both approximately 42.43 km/h. This is found by calculating the sine and cosine of 45 degrees and multiplying them by the magnitude of the velocity (60 km/h).

Now, we can add these vectors together to find the resultant velocity.

Vertical component:
Airplane's velocity (north): 250 km/h
Wind's velocity (north): 42.43 km/h

250 km/h + 42.43 km/h = 292.43 km/h (north)

Horizontal component:
Airplane's velocity (east-west): 0 km/h
Wind's velocity (east): 42.43 km/h

0 km/h + 42.43 km/h = 42.43 km/h (east)

So, the plane's speed relative to the ground is approximately 292.43 km/h and its direction relative to the ground is approximately north 7.17 degrees east.

250 km