An aircraft flying at a steady velocity of 70m/s eastwards at a height of 800m drops a package of supplies. Express the initial velocity of the package as a vector
v = 70 m/s eastward
80
To express the initial velocity of the package as a vector, we need to consider both the magnitude (speed) and direction.
Given that the aircraft is flying at a steady velocity of 70 m/s eastwards, we know that the initial velocity of the package will also have a horizontal component in the east direction.
Let's assume that the horizontal direction is positive eastwards and the vertical direction is positive upwards.
Therefore, the initial velocity of the package can be expressed as a vector:
v_initial = (70 m/s, 0 m/s, 0 m/s)
Where the horizontal component (x-axis) is 70 m/s, and the vertical (y-axis) and depth (z-axis) components are 0 m/s, since the package is dropped vertically downwards. Note that the depth component is zero since we assume it is not moving in the depth direction.
To express the initial velocity of the package as a vector, we need to define its direction and magnitude. In this case, the aircraft is flying eastwards, so we can assume the package is dropped directly downwards from the aircraft.
Since the aircraft is flying at a steady velocity, the horizontal component of its velocity will remain unchanged when the package is dropped. Therefore, the initial velocity of the package will have the same horizontal component as the aircraft's velocity, which is 70 m/s eastwards.
However, when the package is dropped, it will only have a vertical component of velocity. Since the package is dropped, it will have an initial velocity of 0 m/s in the horizontal direction.
Therefore, the initial velocity of the package can be expressed as a vector:
V_initial = (0 m/s, -v_vertical)
Since it's dropping downwards and the height is given as 800m, we can use the equation of motion to find the vertical component of velocity:
v_vertical = sqrt(2gh)
Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the height (800m).
Plugging in the values, we get:
v_vertical = sqrt(2 * 9.8 m/s^2 * 800m) ≈ 392 m/s
So the initial velocity of the package as a vector is:
V_initial = (0 m/s, -392 m/s)