A stunt airplane travels at N 62 E at a constant speed of 400 mi/hr. There is a 20 mi/hr wind blowing doe west. What is the planes actual speed?
Vp = 400mi/h @ 28 Deg.,CCW.
Vw = 20mi/h @ 180 Deg.,CCW.
X = 400*cos28 + 20*cos180 = 333.2 mi/h.
Y = 400*sin28 + 20*sin180 = 187.8mi/h.
V=sqrt((333.2)^2 + (187.8)^2)=382.5 mi/h = Actual speed of plane.
Well, it's a good thing the airplane is a stunt plane because it's going to have to do some fancy tricks to handle that wind! Now, let's calculate the actual speed.
To find the actual speed, we need to consider both the speed of the plane and the speed of the wind. The plane is traveling at a constant speed of 400 mi/hr to the northeast. However, there is a westward wind blowing at 20 mi/hr.
Now, imagine the wind playing a little prank on the plane. It's trying to slow it down and push it in the opposite direction. So, we need to subtract the speed of the wind from the speed of the plane.
Since the wind is coming from the west and the plane is traveling northeast (62 degrees east), we can consider the wind as a side-wind.
Using a bit of trigonometry, we can calculate the component of the wind that is pushing against the plane's direction of travel.
The wind is perpendicular to the plane's direction, forming a right triangle. We can find the length of the side formed by the wind using the formula: wind component = wind speed * cosine(angle).
So, the wind component pushing against the plane is 20 mi/hr * cosine(90° - 62°).
cos(90° - 62°) ≈ cos(28°) ≈ 0.883
Multiplying the wind component by the cosine of the angle gives us:
wind component = 20 mi/hr * 0.883 ≈ 17.66 mi/hr
Now that we have the wind component, we can subtract it from the plane's speed:
actual speed = plane speed - wind component
actual speed = 400 mi/hr - 17.66 mi/hr
actual speed ≈ 382.34 mi/hr
So, the plane's actual speed, accounting for the 20 mi/hr wind pushing against it, is approximately 382.34 mi/hr.
Now that's some serious air acrobatics with a hint of wind humor!
To find the actual speed of the stunt airplane, we need to calculate the resultant speed by considering the wind's effect.
1. Convert the wind speed from miles per hour to east-west component: Since the wind is blowing due west, the east-west component will be -20 mi/hr (negative sign indicates westward direction).
2. Determine the eastward component of the airplane's velocity: The airplane is traveling at a constant speed of 400 mi/hr in the N 62 E direction. To determine the eastward component, we need to find the cosine of the angle between the direction of motion and the east axis (x-axis).
cos(62 degrees) ≈ 0.45399
eastward component = 0.45399 * 400 ≈ 181.6 mi/hr
3. Calculate the resultant velocity: The resultant velocity is the vector sum of the airplane's velocity and the wind's velocity. Since the wind is blowing due west, the north-south component remains unaffected.
eastward component of resultant velocity = eastward component of airplane's velocity + east-west component of wind's velocity
= 181.6 mi/hr - 20 mi/hr (in westward direction)
= 161.6 mi/hr
north-south component of resultant velocity = north-south component of airplane's velocity
= 400 mi/hr
Speed = √(eastward component of resultant velocity)^2 + (north-south component of resultant velocity)^2
= √(161.6)^2 + (400)^2
= √(26131.56 + 160000)
= √186131.56
≈ 431.29 mi/hr
Therefore, the plane's actual speed is approximately 431.29 mi/hr.
To determine the plane's actual speed, we need to consider the effect of the wind on its motion.
Given that the plane is traveling at a constant speed of 400 mi/hr in the direction N 62 E, we can break down its velocity into two components: the northward component (N) and the eastward component (E).
Since the direction is given as N 62 E, we can use trigonometry to find the magnitudes of these components. The northward component can be found by multiplying the plane's speed by the sine of 62 degrees (sin 62°), and the eastward component can be found by multiplying the speed by the cosine of 62 degrees (cos 62°).
Northward component = 400 mi/hr * sin(62°) ≈ 348.19 mi/hr
Eastward component = 400 mi/hr * cos(62°) ≈ 218.28 mi/hr
Now, let's consider the effect of the wind. The wind is blowing due west at a speed of 20 mi/hr. This means that it has a westward component of 20 mi/hr and no northward or eastward component.
To find the actual speed of the plane, we need to combine the plane's velocity components with the wind's component. Since the wind is blowing westward, we will subtract its speed component from the eastward component of the plane.
Actual speed = sqrt((eastward component - westward component)^2 + northward component^2)
Actual speed = sqrt((218.28 mi/hr - 20 mi/hr)^2 + (348.19 mi/hr)^2)
Actual speed = sqrt(198.28^2 + 348.19^2)
Actual speed ≈ 400.07 mi/hr
Therefore, the plane's actual speed, considering the effect of the wind, is approximately 400.07 mi/hr.