A jetliner with an airspeed of 1000 kmlhr sets out on a 1500 km flight due south. To maintain
a southward direction, however, the plane must be pointed fifteen degrees west of south. Ifthe
flight takes 1 00 min, what is the wind velocity?
if the wind speed is s, set up the x- and y-components of the vectors:
first, work with velocity vectors only.
1500km in 100 min = 900km/hr
so, using the law of cosines,
s^2 = 1000^2 + 900^2 - 2(100)(90)cos30°
s = 1340 km/hr
To determine the wind velocity, we need to first figure out the ground speed of the jetliner. The ground speed is the actual speed of the jetliner relative to the ground and is influenced by the airspeed and the wind velocity.
Here's how we can find the ground speed and eventually determine the wind velocity:
Step 1: Calculate the distance covered by the jetliner in 100 minutes.
- The distance can be calculated using the formula: distance = speed × time.
- In this case, the speed is the airspeed of the jetliner, which is 1000 km/hr.
- Convert the time from minutes to hours: 100 min ÷ 60 min/hr = 1.67 hrs.
- Calculate the distance: distance = 1000 km/hr × 1.67 hrs = 1670 km.
Step 2: Determine the southward component of the ground speed.
- Since the plane must be pointed 15 degrees west of south, we need to find the southward component of the ground speed.
- This can be calculated using the formula: southward component = ground speed × cos(angle).
- The angle is 15 degrees, and we need to convert it to radians: angle (in radians) = 15 degrees × π/180 ≈ 0.2618 radians.
- Calculate the southward component: southward component = ground speed × cos(0.2618).
Step 3: Determine the westward component of the ground speed.
- Since the plane must be pointed 15 degrees west of south, we need to find the westward component of the ground speed.
- This can be calculated using the formula: westward component = ground speed × sin(angle).
- Calculate the westward component: westward component = ground speed × sin(0.2618).
Step 4: Determine the resultant ground speed.
- Since the plane is flying due south, the resultant ground speed will only have a southward component.
- The resultant ground speed can be calculated using the Pythagorean theorem: resultant ground speed = √(southward component^2 + westward component^2).
Step 5: Determine the wind velocity.
- The wind velocity is the difference between the airspeed and the ground speed.
- Calculate the wind velocity by subtracting the airspeed from the resultant ground speed.
By following these steps, we can find the wind velocity based on the given information.