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Vertical motion describes how the suitcase falls after it leaves the airplane. Since gravity is the only force acting on it vertically, the suitcase undergoes constant acceleration downwards at approximately 9.81 m/s².
This is known as free fall. To calculate how long it takes the suitcase to reach the ground, we use formulas from kinematics.
This involves the initial vertical velocity and the height from which it was dropped.

• Initial vertical velocity is calculated from the velocity of the plane and the angle at which it flies.
• The height is the initial vertical distance between the plane and the ground.

For this suitcase, the equation involves replacing into the quadratic formula to solve for time. This calculation results in a time of about 8.60 seconds.

While the suitcase is falling vertically, its horizontal motion is straightforward. Since there are no horizontal forces acting on the suitcase (ignoring air resistance), it maintains a constant velocity. In our exercise, this velocity is derived from the horizontal component of the airplane's initial velocity.
The horizontal distance traveled only depends on this constant horizontal velocity and the duration it takes for the suitcase to fall, also known as the time of flight.

• The longer the time of flight, the farther the suitcase will travel horizontally.
• With a calculated horizontal speed of about 82.83 m/s, the suitcase covers more distance over more time.

Hence, the overall motion is the combination of falling vertically under gravity and moving horizontally at a steady pace.

To accurately determine the movement of the suitcase upon release, it is crucial to break down its initial velocity into vertical and horizontal components. This separation of components is foundational in understanding projectile motion.
Let's explore how to calculate these components:

• **Horizontal Component**: This is calculated using the formula \( v_x = v \cos \theta \), where \( v \) is the initial velocity of the plane, and \( \theta \) is the angle of projection. For the given problem, this component is approximately 82.83 m/s.
• **Vertical Component**: Similarly, this is determined using \( v_y = v \sin \theta \). For our suitcase, it turns out to be around 35.15 m/s.

By breaking the velocity into these two components, we can separately analyze how the suitcase moves vertically and horizontally.

The time of flight is the time the suitcase spends in the air from the moment it drops until it hits the ground. In the case of the suitcase, it's affected primarily by the vertical motion under gravity. Calculating this requires solving a quadratic equation obtained from the vertical motion formula:
\[ h = v_y t + \frac{1}{2} g t^2 \] Here, \( h \) is the initial height (114 meters), \( v_y \) is the vertical component of initial velocity, and \( g \) is the acceleration due to gravity.
The quadratic formula is then used to solve for \( t \), resulting in approximately 8.60 seconds.

• During this time, the suitcase falls and covers a distance horizontally at the initial horizontal velocity.
• Hence, understanding the time of flight helps in predicting both vertical landing point and horizontal range.

Consequently, the suitcase lands far from its initial drop point due to its steady horizontal speed over 8.60 seconds.