A small plane can travel at 200 km/h in still air. If a 50.0 km/h wind is coming from the east, determine the ground velocity of the plane if the pilot keeps the plane pointed [N].
This is just pythagorean theorem/trig.
Basically the plane's velocity and the wind velocity are both "legs" of a right triangle. You apply the pythagorean theorem to find the ground velocity.
square root of (200^2+50^2) = 206 km/h. Try it!
Also, if you need to find the degree north west the plane is going, you can use trigonometry: (finding the inverse tangent)
tan^-1(50/200) = 14 degrees.
Note that the answers are all approximate.
To determine the ground velocity of the plane, we need to consider the effect of the wind. Since the wind is coming from the east, it will affect the plane's velocity when moving north.
The ground velocity of the plane can be calculated using vector addition. We can break down the plane's velocity into two components: the northward component and the eastward component.
Given that the plane can travel at a speed of 200 km/h in still air, and there is a 50.0 km/h wind coming from the east, we can calculate the northward component of the ground velocity:
Northward component = Plane's speed * sine of the angle between the plane's track and the north direction
Since the plane is pointed north, the angle between the plane's track and the north direction is 0 degrees.
Northward component = 200 km/h * sin(0°)
Northward component = 0 km/h
The northward component of the ground velocity is zero, which means the wind does not affect the plane's northward motion.
Next, let's calculate the eastward component of the ground velocity:
Eastward component = Plane's speed * cosine of the angle between the plane's track and the north direction
Since the plane is pointed north, the angle between the plane's track and the north direction is 90 degrees.
Eastward component = 200 km/h * cos(90°)
Eastward component = 0 km/h
The eastward component of the ground velocity is zero, which means the wind cancels out the plane's eastward motion.
Therefore, the ground velocity of the plane, when the pilot keeps the plane pointed north, is 0 km/h. The wind completely negates the plane's forward motion in this case.
To determine the ground velocity of the plane, we need to consider the effect of the wind on its motion.
First, let's define the variables:
- Plane speed in still air: 200 km/h
- Wind speed: 50.0 km/h (coming from the east)
- North direction: We'll assume it is perpendicular to the east-west direction
To find the ground velocity, we can use vector addition. We'll consider the plane's velocity relative to the air and the wind's velocity as separate vectors, and then add them together.
Step 1: Resolve the velocities into components
The velocity of the plane relative to the air is towards the north, as it is pointed north. Therefore, its northward component is 200 km/h.
The wind is coming from the east, which means its vector points west. Therefore, its western (negative) component is 50.0 km/h.
Step 2: Add the components
To find the ground velocity, we need to add the northward component of the plane's velocity and the western component of the wind's velocity, as they are perpendicular to each other.
Northward component of the plane's velocity = 200 km/h
Western component of the wind's velocity = -50.0 km/h
Adding these components gives us the ground velocity of the plane.
Step 3: Find the magnitude and direction of the ground velocity vector
To find the magnitude (speed) of the ground velocity, we can use the Pythagorean theorem because the components form a right triangle.
Magnitude of the ground velocity = square root of (northward component^2 + western component^2)
Magnitude of the ground velocity = square root of (200^2 + (-50)^2)
Magnitude of the ground velocity ≈ 208.21 km/h (rounded to two decimal places)
To find the direction of the ground velocity, we can use inverse tangent (tan^(-1)) of the northward component divided by the western component:
Direction of the ground velocity = tan^(-1)(northward component / western component)
Direction of the ground velocity = tan^(-1)(200 / -50)
Direction of the ground velocity ≈ -75.96° (rounded to two decimal places)
Therefore, the ground velocity of the plane, if the pilot keeps the plane pointed north, is approximately 208.21 km/h to the north at an angle of -75.96° (measured counterclockwise from east).