Problem 91 - Fundamentals of Physics

Question

\(A\) rock is dropped from a \(100-\mathrm{m}\) -high cliff. How long does it take to fall (a) the first \(50 \mathrm{~m}\) and (b) the second \(50 \mathrm{~m}\) ?

Step-by-Step Solution

Answer

(a) 3.19 s (b) 1.33 s

Step 1: Understanding the Problem

We need to find the time it takes for a rock to fall two separate distances: the first 50 meters and the second 50 meters from a total height of 100 meters. The rock is falling under the influence of gravity, so we will apply the equations of motion for uniformly accelerated objects.

Step 1: Write the Equation of Motion

The equation of motion to calculate the distance fallen under gravity is given by \[ s = ut + \frac{1}{2}gt^2 \]where:- \( s \) is the distance fallen,- \( u \) is the initial velocity,- \( g \) is the acceleration due to gravity (approximately \(9.8\, \mathrm{m/s^2}\)),- \( t \) is the time taken. For this problem, since the rock is initially at rest, \( u = 0 \).

Step 2: Calculate Time for First 50 m

Using the equation \( s = \frac{1}{2}gt^2 \), we set \( s = 50 \) m and solve for \( t \):\[50 = \frac{1}{2} \times 9.8 \times t^2\]\[50 = 4.9t^2\]\[t^2 = 10.2\]\[t = \sqrt{10.2} \approx 3.19 \, \mathrm{s}\]Thus, it takes approximately 3.19 seconds to fall the first 50 meters.

Step 3: Calculate Time for the Total 100 m

Now, calculate the total time to fall 100 meters using the same equation format:\[100 = \frac{1}{2} \times 9.8 \times T^2\]\[100 = 4.9T^2\]\[T^2 = 20.4\]\[T = \sqrt{20.4} \approx 4.52 \, \mathrm{s}\]The total time for 100 meters is approximately 4.52 seconds.

Step 4: Calculate Time for Second 50 m

The time to fall the second 50 meters is the difference between the total time for 100 meters and the time for the first 50 meters:\[4.52\, \mathrm{s} - 3.19\, \mathrm{s} \approx 1.33 \, \mathrm{s}\]Thus, it takes approximately 1.33 seconds to fall the second 50 meters.

Key Concepts

Understanding Free Fall

In physics, free fall occurs when the only force acting on an object is gravity. This means that any resistance, like air resistance, is neglected, allowing the object to accelerate downward purely under the influence of gravity. In a free-fall scenario, objects are initially at rest, meaning their starting velocity is zero. This aligns perfectly with motion equations where initial velocity is denoted as zero in calculations.

Let's consider why free fall is essential in our problem. When a rock falls off a cliff, it undergoes free fall, making it an ideal candidate for using motion equations.

• No other forces affect the rock besides gravity.
• We can determine the time taken to fall a certain distance easily.

Understanding these principles helps greatly with solving equations related to distance, velocity, and time when the object is under the influence of gravity.

The Role of Gravity

Gravity is the key player in free fall motion and directly influences how objects move when they are dropped. It is the acceleration that causes objects to increase their speed as they descend. On Earth, gravity accelerates all objects at approximately \(9.8 \, \mathrm{m/s^2}\).This constant acceleration means:

• Objects will gain speed as they fall towards the ground.
• This increment in speed continues until the object hits the ground or is acted upon by another force, like air resistance.

Gravity allows for the use of motion equations to predict how long it takes for an object to fall a given distance. The motion equations simplify calculations by considering gravity's constant acceleration, allowing students to solve problems about falling objects more easily.

Time Calculation in Free Fall

Calculating the time it takes for an object to fall a specific distance during free fall involves using motion equations where time, distance, and acceleration are connected. In these calculations, we focus on the equation \[ s = \frac{1}{2}gt^2 \].Here's a step-by-step breakdown of how the formula works:

• \( s \) represents the distance the object has fallen, which is given in the problem.
• \( g \) stands for the acceleration due to gravity, set as \(9.8 \, \mathrm{m/s^2}\).
• \( t \) is the time, which is what we need to find.

By solving for \( t \), you discover how long it takes for the object to fall a specific distance.

In the given exercise, calculating the time for different distances can be done by first considering the total time it takes to reach a point and then the incremental difference to get from one point to another. This method effectively splits a larger free-fall scenario into smaller, more manageable parts for detailed analysis.