It says, A plane travels 400 mi/h relative to the ground. There is a 25 mi/h wind blowing from the west to the east. Find the plane's resultant velocity as it:
a. travels west:
b. travels east:
c. travels north:
d. The plane's final destination is 1250 miles north. If the plane heads directly north, how far off course will the plane be blown?
e. What direction must the plane move toward in order to head straight north? (preAP, find angle)
f. What would the planes speed be, relative to the ground, if it were to head straight north?
a. V = 400 - 25 = 375 mi/h.
b. V = 400 + 25 = 425 mi/h.
c. X = hor = 25 mi/h.
Y = ver = 400 mi/h.
tanA = Y/X = 400/25 = 16,
A = 86.4 deg CCW.
d. B = 90 - 86.4 = 3.6 deg.
e. 90 + 3.6 = 93.6 deg.
f. V = 400 / sin86.4 = 400.8 mi/h.
To solve these questions, we need to consider the velocity of the plane relative to the ground, taking into account the wind speed and direction. The plane's resultant velocity can be found by adding the plane's velocity relative to the ground and the velocity of the wind.
a. When the plane travels west, the wind is blowing from the west to the east. Therefore, the wind will oppose the plane's motion, reducing its ground speed. To find the plane's resultant velocity when traveling west, subtract the wind velocity from the plane's velocity:
Resultant velocity = Plane velocity - Wind velocity
= 400 mi/h - 25 mi/h (since the wind is blowing from west to east)
= 375 mi/h
So, when the plane travels west, its resultant velocity is 375 mi/h.
b. When the plane travels east, the wind is blowing from the west to the east. This time, the wind will assist the plane's motion, increasing its ground speed. To find the plane's resultant velocity when traveling east, add the wind velocity to the plane's velocity:
Resultant velocity = Plane velocity + Wind velocity
= 400 mi/h + 25 mi/h (since the wind is blowing from west to east)
= 425 mi/h
So, when the plane travels east, its resultant velocity is 425 mi/h.
c. When the plane travels north, the direction of the wind does not affect its northward motion. Therefore, the plane's resultant velocity when traveling north is the same as its original velocity relative to the ground, which is 400 mi/h.
d. If the plane's final destination is 1250 miles north, but it heads directly north, the wind will blow the plane off course. To calculate how far off course the plane will be blown, we need to determine the time it takes the plane to reach its destination and multiply it by the wind's horizontal (east-west) blowing speed.
Time taken = Distance / Velocity
= 1250 miles / 400 mi/h
= 3.125 hours
Distance off course = Time taken * Wind velocity
= 3.125 hours * 25 mi/h (since the wind is blowing from west to east)
= 78.125 miles
So, the plane will be blown approximately 78.125 miles off course.
e. To head straight north, the plane needs to counteract the wind blowing from west to east. This means the plane will need to steer slightly to the west in order to compensate for the wind. To find the angle at which the plane must move, we can use trigonometry.
Angle = arctan( Wind velocity / Plane velocity )
= arctan( 25 mi/h / 400 mi/h )
= arctan( 0.0625 )
≈ 3.573 degrees
Therefore, the plane must move at an angle of approximately 3.573 degrees to the west in order to head straight north.
f. If the plane were to head straight north, its speed relative to the ground would be the same as its original velocity relative to the ground, which is 400 mi/h. Thus, the plane's speed relative to the ground would remain 400 mi/h when heading straight north.