An airplane, traveling 90 degrees, at 100 meters per second is blown towards 180 degrees at 50 meters per second. What is the resultant velocity and direction?
To find the resultant velocity and direction, we need to calculate the vector sum of the airplane's velocity and the velocity of the wind.
1. Start by breaking down the airplane's velocity and the wind's velocity into their horizontal and vertical components.
Given:
- Airplane velocity = 100 m/s at 90 degrees (relative to the positive x-axis)
- Wind velocity = 50 m/s at 180 degrees
Airplane velocity components:
- Horizontal component = 100 m/s * cos(90 degrees) = 0 m/s
- Vertical component = 100 m/s * sin(90 degrees) = 100 m/s
Wind velocity components:
- Horizontal component = 50 m/s * cos(180 degrees) = -50 m/s
- Vertical component = 50 m/s * sin(180 degrees) = 0 m/s
2. Add the horizontal components and vertical components separately to get the resultant vectors.
Resultant horizontal velocity = Airplane's horizontal component + Wind's horizontal component
= 0 m/s + (-50 m/s) = -50 m/s
Resultant vertical velocity = Airplane's vertical component + Wind's vertical component
= 100 m/s + 0 m/s = 100 m/s
3. Use the Pythagorean theorem to find the magnitude of the resultant velocity.
Resultant velocity = √(Resultant horizontal velocity^2 + Resultant vertical velocity^2)
= √((-50 m/s)^2 + (100 m/s)^2)
= √(2500 m^2/s^2 + 10000 m^2/s^2)
= √12500 m^2/s^2
≈ 111.80 m/s
4. Find the direction of the resultant velocity using the inverse tangent function.
Resultant direction = tan^(-1)(Resultant vertical velocity / Resultant horizontal velocity)
= tan^(-1)(100 m/s / -50 m/s)
= tan^(-1)(-2)
≈ -63.43 degrees
Note: The negative sign indicates that the resultant velocity is in the opposite direction of the positive x-axis.
Therefore, the resultant velocity is approximately 111.80 m/s at -63.43 degrees (relative to the positive x-axis).