An airplane with a speed of 46.4 m/s is climbing upward at an angle of 44° counterclockwise from the positive x axis. When the plane's altitude is 700 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.
(b) Determine the angle of the velocity vector of the package just before impact.
To solve this problem, we can break it down into components: the horizontal and vertical components of the airplane's motion, as well as the horizontal and vertical components of the package's motion.
(a) To calculate the distance along the ground, we need to find the time it takes for the package to hit the ground after being released.
Step 1: Find the vertical component of the airplane's velocity.
Given: Speed of the airplane = 46.4 m/s, Angle of climb = 44°
Use trigonometry to find the vertical component:
Vertical velocity = Speed * sin(angle)
Vertical velocity = 46.4 m/s * sin(44°)
Step 2: Find the time it takes for the package to hit the ground.
Given: Initial altitude = 700 m, Acceleration due to gravity = 9.8 m/s²
Use kinematic equations to find the time:
Using the equation s = ut + 0.5 * a * t², where s is displacement, u is initial velocity, a is acceleration, and t is time.
Setting s = -700 m (negative because the package is moving downward), u = vertical velocity, and a = -9.8 m/s² (negative because acceleration is opposite to motion):
-700 m = (vertical velocity) * t + 0.5 * (-9.8 m/s²) * t²
This is a quadratic equation that can be solved to find t.
Step 3: Calculate the horizontal distance traveled by the package.
Given: Time = t, Horizontal velocity = Speed * cos(angle)
Horizontal distance = Horizontal velocity * time
(b) To determine the angle of the velocity vector of the package just before impact, we can find the angle of its velocity vector by using trigonometry.
Step 1: Find the horizontal component of the package's velocity.
Given: Horizontal velocity = Horizontal velocity of the airplane
Step 2: Find the vertical component of the package's velocity.
Given: Time = t, Vertical velocity = -9.8 m/s² * t (since the package is moving downward under the influence of gravity)
Vertical distance = Vertical velocity * time
Step 3: Find the resultant velocity of the package using the Pythagorean theorem.
Resultant velocity = √(horizontal velocity² + vertical velocity²)
Step 4: Find the angle of the velocity vector.
Given: Resultant velocity = ??, Horizontal velocity = ??
Use trigonometry to find the angle:
Angle = arctan(vertical velocity / horizontal velocity)
Following these steps should allow you to calculate the required values.