Problem 48 - College Physics

Question

Suppose that you drop a marble from the top of the Burj Khalifa building in Dubai, which is about \(830 \mathrm{~m}\) tall. If you ignore air resistance, (a) how long will it take for the marble to hit the ground? (b) How fast will it be moving just before it hits?

Step-by-Step Solution

Answer

(a) 13 seconds, (b) 127.4 m/s.

Step 1: Identifying the Known Values

We know the height from which the marble is dropped, which is \( h = 830 \) m. The initial velocity \( u = 0 \) m/s since it is dropped. The acceleration due to gravity \( g = 9.8 \) m/s² (assuming no air resistance).

Step 2: Calculating the Time to Hit the Ground

We will use the kinematic equation \( h = ut + \frac{1}{2}gt^2 \) where \( u = 0 \) m/s. This simplifies to \( h = \frac{1}{2}gt^2 \). Substituting the known values: \( 830 = \frac{1}{2} \times 9.8 \times t^2 \). Solving for \( t \), we get \[ t^2 = \frac{830 \times 2}{9.8} \ \approx 169.39 \]. Thus, \( t \approx 13 \) seconds.

Step 3: Calculating the Final Velocity

Use the kinematic equation \( v = u + gt \). Since the initial velocity \( u = 0 \), the equation simplifies to \( v = gt \). Substituting the known values: \( v = 9.8 \times 13 \ \approx 127.4 \) m/s.

Key Concepts

Acceleration Due to Gravity

The acceleration due to gravity, often represented by the symbol \( g \), is the rate at which an object accelerates when it is falling freely toward Earth. On our planet, this value is approximately \( 9.8 \text{ m/s}^2 \). This means that every second, the velocity of a freely falling object increases by \( 9.8 \text{ m/s} \).
When calculating problems involving objects under free fall, we generally assume that air resistance is negligible. This simplifies our calculations and allows us to rely on the constant value of gravity. This concept is crucial in understanding the motion of objects not only on Earth but also helps in studying celestial bodies. Many physics problems start by identifying this constant to predict how objects move under the influence of gravity.

Free Fall

Free fall is a special kind of motion that occurs when only gravity is acting on an object. When an object is in free fall, it does not encounter air resistance or external forces.

Imagine dropping a marble from a tall building like the Burj Khalifa. At the moment it leaves your hand, it's in free fall if air resistance is ignored. In this condition:

• The only force acting on it is the gravitational pull.
• Its initial velocity is zero if you simply let go of it.
• It accelerates continuously at the rate of gravity \( g \), which is \( 9.8 \text{ m/s}^2 \) on Earth.

Understanding the principles of free fall is essential in physics because it provides a basis for analyzing more complex motions when other forces come into play.

Motion Equations

To analyze the motion of objects, such as a marble being dropped from a great height, we utilize motion equations, specifically the kinematic equations. These help us describe motion mathematically without needing to delve into the reasons behind it.
One key equation is used to determine the time it takes an object to reach the ground, and another is used to find its final velocity just before impact. Here's how they both look:

• To find time \( t \), we use \( h = \frac{1}{2}gt^2 \). Solving for \( t \) gives us the time it takes for a free-falling object to hit the ground from a known height \( h \).
• To find the final velocity \( v \) just before it hits the ground, we use \( v = gt \).

The simplicity of these equations allows us to solve complex motion problems with minimal input data. By measuring the height and using the constant acceleration due to gravity, predicting the outcome of a free fall experiment becomes straightforward.