Problem 3 - Applied Physics

Question

A piece of shattered glass falls from the 8 2nd floor of a building, \(27 \overline{0} \mathrm{~m}\) above the ground. What is the velocity of the glass when it hits the ground, if air resistance is disregarded?

Step-by-Step Solution

Answer

The velocity is approximately 23.01 m/s downward.

Step 1: Identify Given Values

The height from which the piece of glass falls is given as 27.0 meters. The initial velocity (\(v_i\)) is 0 since the glass starts from rest. The acceleration due to gravity (\(g\)) is approximately 9.81 m/s² downward.

Step 2: Use the Kinematic Equation

The kinematic equation relating initial velocity, final velocity, acceleration, and displacement is\(v_f^2 = v_i^2 + 2ad\), where\(v_f\) is the final velocity,\(v_i\) is the initial velocity (0 m/s),\(a\) is the acceleration (9.81 m/s²), and\(d\) is the displacement (27.0 m).

Step 3: Substitute the Values

Substitute the known values into the equation: \(v_f^2 = 0 + 2 \times 9.81 \times 27\).

Step 4: Calculate Final Velocity

Calculate\(v_f^2 = 2 \times 9.81 \times 27 = 529.62\). Take the square root to find\(v_f = \sqrt{529.62} \approx 23.01\) m/s.

Step 5: Interpret the Result

The velocity of the glass when it hits the ground is approximately 23.01 m/s downward.

Key Concepts

Projectile Motion

Projectile motion refers to the movement of an object that is thrown or projected into the air and is subject to gravity. This kind of motion involves two components: horizontal motion and vertical motion. In projectile motion, we often see:

• The object follows a curved trajectory, often a parabola.
• Horizontal velocity remains constant if air resistance is neglected.
• Vertical velocity changes at a constant rate due to gravity.

In the case of the falling glass from the exercise, it mainly demonstrates vertical projectile motion since the initial horizontal velocity is zero. Unlike typical projectile motion where there is both horizontal and vertical components, here, the motion is purely vertical. The glass starts from rest and gains speed as it falls towards the ground due to gravity. Understanding projectile motion fundamentally requires breaking down these components to solve for various unknowns like time, range, and final velocities.

Free Fall

Free fall is a specific type of projectile motion in which an object moves under the influence of gravity alone. In free fall, the object doesn't encounter any resistance, such as air resistance, which makes calculations simpler. Key points to remember about free fall include:

• Initial velocity can be zero if the object starts from rest, as with the glass.
• The only acceleration acting on the object is due to gravity, which is approximately 9.81 m/s² downwards.
• All objects in free fall, irrespective of their masses, accelerate at the same rate if we ignore air resistance.

In our exercise, the glass exhibits free fall as it descends from the building's height. We calculate its velocity upon reaching the ground by applying kinematic equations that consider height, initial velocity (zero in this case), and acceleration due to gravity. Observing free fall in this way allows us to predict and calculate final velocities and other parameters when dealing with similar physics problems.

Acceleration Due to Gravity

Acceleration due to gravity, denoted by the symbol \( g \), is a constant value on Earth's surface that measures the force exerted by the Earth on an object. In most physics calculations, it is approximated as 9.81 m/s². Let's explore some fundamental aspects:

• \( g \) causes all objects to accelerate towards the Earth when in free fall.
• This value is primarily uniform over the Earth's surface and only slightly varies depending on altitude or geographical location.
• Acceleration is always directed downwards, towards the center of the Earth.

When solving for cases of free fall, including the example of the shattered glass falling, \( g \) becomes an essential component of our kinematic equations. Specifically, we use \( g \) in the formula \( v_f^2 = v_i^2 + 2ad \) to determine how velocity changes over time and distance. Recognizing the role of gravity is crucial in solving kinematics problems that involve falling objects or parabolic trajectories.