Certainly, I'd be happy to explain the step-by-step solution for this scenario.
1. Understand the Directions and Components:
- The plane's airspeed is 350 mph in the direction [tex]\( N 40^{\circ} E \)[/tex].
- A wind is blowing from the south at 50 mph, which means towards the north.
2. Resolve the Plane's Airspeed into Components:
- The plane's direction [tex]\( N 40^{\circ} E \)[/tex] can be broken down into components along the east (x-direction) and north (y-direction) axes.
- The angle of [tex]\( 40^{\circ} \)[/tex] is measured from the north towards the east. So:
- The eastward component (x) is [tex]\( 350 \cos 40^{\circ} \)[/tex].
- The northward component (y) is [tex]\( 350 \sin 40^{\circ} \)[/tex].
3. Include the Wind’s Impact:
- Since the wind is blowing from the south, it will add directly to the northward (y) component of the plane's velocity.
- The eastward component (x) remains unchanged.
4. Calculate the Components:
- Eastward (x) component of the plane's airspeed:
- [tex]\( x = 350 \cos 40^{\circ} \approx 268.12 \, \text{mph} \)[/tex].
- Northward (y) component of the plane's airspeed:
- [tex]\( y = 350 \sin 40^{\circ} \approx 224.98 \, \text{mph} \)[/tex].
5. Resultant Components with Wind Effect:
- Resultant eastward (x) component remains the same: [tex]\( 268.12 \, \text{mph} \)[/tex].
- Resultant northward (y) component with wind added: [tex]\( 224.98 \, \text{mph} + 50 \, \text{mph} = 274.98 \, \text{mph} \)[/tex].
6. Calculate the Ground Speed:
- Use the Pythagorean theorem to find the ground speed [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt{ (268.12)^2 + (274.98)^2 } \approx 384.05 \, \text{mph}
\][/tex]
Hence, the components of the plane's airspeed are approximately:
- Eastward component: [tex]\( 268.12 \, \text{mph} \)[/tex].
- Northward component: [tex]\( 224.98 \, \text{mph} \)[/tex].
The resultant components after accounting for the wind are:
- Eastward component: [tex]\( 268.12 \, \text{mph} \)[/tex].
- Northward component: [tex]\( 274.98 \, \text{mph} \)[/tex].
The ground speed of the plane is approximately [tex]\( 384.05 \, \text{mph} \)[/tex].
This detailed decomposition helps visualize how the wind affects the plane's airspeed and leads to the calculation of the new ground speed.
