Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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.
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.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
