You are in a hot-air balloon that, relative to the ground, has a velocity of 5.40 m/s in a direction due east. You see a hawk moving directly away from the balloon in a direction due north. The speed of the hawk relative to you is 1.05 m/s. What are (a) the magnitude and (b)direction of the hawk's velocity relative to the ground? Express the directional angle relative to due east.
To determine the magnitude and direction of the hawk's velocity relative to the ground, we need to consider the velocities of both the hot-air balloon and the hawk.
Let's break down the information given:
- The hot-air balloon's velocity relative to the ground is 5.40 m/s due east. We can denote this as v_balloon.
- You see the hawk moving directly away from the balloon in a direction due north. The speed of the hawk relative to you is 1.05 m/s. We can denote this as v_hawk_relative.
To find the hawk's velocity relative to the ground, we'll use vector addition. The hawk's velocity relative to you (v_hawk_relative) and the balloon's velocity relative to the ground (v_balloon) can be added as vectors to find the resultant velocity of the hawk relative to the ground (v_hawk_ground).
(a) To find the magnitude of the hawk's velocity relative to the ground, we'll use the Pythagorean theorem:
magnitude = sqrt(v_balloon^2 + v_hawk_relative^2)
magnitude = sqrt((5.40 m/s)^2 + (1.05 m/s)^2)
magnitude ≈ sqrt(29.16 m^2/s^2 + 1.1025 m^2/s^2)
magnitude ≈ sqrt(30.2625 m^2/s^2)
magnitude ≈ 5.5 m/s
Therefore, the magnitude of the hawk's velocity relative to the ground is approximately 5.5 m/s.
(b) To find the direction of the hawk's velocity relative to the ground, we'll use the inverse tangent function (arctan).
direction = arctan(v_hawk_relative / v_balloon)
direction = arctan(1.05 m/s / 5.40 m/s)
direction ≈ arctan(0.1944)
direction ≈ 10.99°
Therefore, the direction of the hawk's velocity relative to the ground is approximately 10.99° relative to due east.
So, the answer is:
(a) The magnitude of the hawk's velocity relative to the ground is approximately 5.5 m/s.
(b) The direction of the hawk's velocity relative to the ground is approximately 10.99° relative to due east.