An airplane with a speed of 27.4 m/s is climbing upward at an angle of 48° counterclockwise from the positive x axis. When the plane's altitude is 860 m the pilot releases a package.
(a) Calculate the distance along the ground, measured from a point directly beneath the point of release, to where the package hits the earth.
Determine the angle of the velocity vector of the package just before impact.
Vo = 27.4m/s[48o]
Xo = 27.4*Cos48 = 18.3 m/s.
Yo = 27.4*sin48 = 20.4 m/s.
h = 0.5g*t^2 = 860 m.
4.9*t^2 = 860
t^2 = 175.5
Tf = 13.2 s. = Fall time.
Dx = Xo*Tf = 18.3m/s * 13.2s = 242.4 m.
b. Y = Yo + g*t = 0 + 9.8*13.2 = 129.4
m/s.
Tan A = Y/Xo = 129.4/18.3 = 7.07104
A = 82o.
To calculate the distance along the ground to where the package hits the earth, you need to find the horizontal component of the airplane's velocity.
The horizontal component of the velocity can be found by multiplying the airplane's speed (27.4 m/s) by the cosine of the angle of climb (48°):
Horizontal component of velocity = 27.4 m/s * cos(48°)
To find the angle of the velocity vector of the package just before impact, you can use the concept of relative velocity. The package has the same horizontal velocity as the airplane, but its vertical velocity is affected by gravity.
The time it takes for the package to hit the ground can be found by solving the equation:
Vertical distance = Initial vertical velocity * time - 0.5 * acceleration due to gravity * time^2
In this case, the initial vertical velocity is the vertical component of the airplane's velocity, which is given by 27.4 m/s * sin(48°). Also, the acceleration due to gravity is approximately 9.8 m/s^2.
With the vertical distance (860 m) known, you can solve for the time it takes for the package to hit the ground.
Then, you can use this time along with the horizontal velocity to calculate the distance along the ground traveled by the package:
Distance along the ground = Horizontal component of velocity * time
Finally, to determine the angle of the velocity vector just before impact, you need to find the tangent of the angle:
Tangent of angle = Vertical component of velocity / Horizontal component of velocity
Now, you can plug in the values and calculate the answers.