91Ó°ÊÓ

Vertical motion refers to the movement of an object along a vertical path. In the context of our problem, this involves the rock being thrown upward against gravity from the building's roof. When dealing with vertical motion:

• We consider an object's initial and final positions.
• The initial velocity at which it is launched.
• The acceleration due to gravity, which is always acting downward at approximately 9.8 m/s² on Earth.

For an object thrown upwards in vertical motion, it will first move up until its velocity becomes zero. Then, it reverses direction and starts accelerating downwards due to gravity. Utilizing equations of motion helps us determine various unknowns such as time of flight (when the object hits the ground) and velocity at any point in its trajectory.

Quadratic equations often arise in vertical motion problems because the motion's mathematical model is quadratic in nature. The quadratic equation is a second-degree polynomial and can be written as: \[ ax^2 + bx + c = 0 \] Here, the coefficients \( a \), \( b \), and \( c \) represent real numbers, and \( x \) is the variable. Quadratic equations can have two solutions, calculated using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where:

• \( b^2 - 4ac \) is the discriminant.
• If the discriminant is positive, two distinct real roots exist.
• If zero, there's exactly one real root.
• If negative, there are no real roots (implying complex solutions).

For our exercise, solving the quadratic equation helps determine the time when the rock reaches the ground, as height (position) is a quadratic function of time in such scenarios.

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. The fundamental kinematic equations help to analyze motion under uniform acceleration or deceleration. For vertical motion, we use three primary kinematic equations:

• Displacement: \( s = ut + \frac{1}{2}at^2 \)
• Final velocity: \( v = u + at \)
• Velocity squared: \( v^2 = u^2 + 2as \)

These equations are invaluable as they relate variables like time \( t \), initial velocity \( u \), final velocity \( v \), displacement \( s \), and acceleration \( a \). In our given problem, these formulas allow us to solve for the time taken for the rock to hit the ground and its velocity at impact.

Free fall specifically refers to motion where gravity is the only force acting on an object. In these scenarios, air resistance is negligible, meaning the only acceleration experienced is due to gravity, approximately 9.8 m/s², directed downwards.
For our exercise, we assume free fall conditions once the rock is thrown upwards and then descends. This simplifies calculations since the only force we consider is gravity. The object's initial upward velocity will reduce to zero as it heads skyward, and during its descent, it accelerates under the influence of gravity alone.
Under free fall:

• Objects gain speed in a downward direction at a constant rate.
• The final velocity just before impact can be determined using kinematic equations, taking into account both the initial upward motion and subsequent fall back to the ground.
• The total time of flight can also be computed by analyzing the upward and downward journey together.

Understanding free fall and its effects help predict an object’s position and velocity at any point in time during its fall.