Problem 17 - An Introduction to Physical Science

Question

shows the Hoover Dam Bridge over the Colorado River at a height of \(274 \mathrm{m}\) If a heavy object is dropped from the bridge, how much time passes before the object makes a splash?

Step-by-Step Solution

Answer

The object makes a splash after about 7.47 seconds.

Step 1: Understanding the Problem

We are asked to find the time it takes for an object to fall from the Hoover Dam Bridge, which is 274 meters above the water, to the point where it hits the water. This is a free-fall physics problem.

Step 2: Identify Known Variables

We know that the height ( 274 ext{ m}) is the distance the object falls and the acceleration due to gravity ( 9.81 ext{ m/s}^2) is constant. The initial velocity ( 0 ext{ m/s}) since the object is dropped.

Step 3: Use the Free Fall Formula

The formula to use is \( h = \frac{1}{2} g t^2 \), where \( h \) is the height, \( g \) is the gravitational acceleration, and \( t \) is the time in seconds.

Step 4: Plug in Known Values

Substitute the known values into the formula: \( 274 = \frac{1}{2} \times 9.81 \times t^2 \).

Step 5: Solve for Time \( t \)

First, simplify the equation: \( 274 = 4.905 \times t^2 \). Then, solve for \( t^2 \): \( t^2 = \frac{274}{4.905} \).

Step 6: Calculate \( t \)

Compute \( t^2 = 55.87 \) and find \( t \) by taking the square root: \( t \approx \sqrt{55.87} \approx 7.47 \).

Step 7: Conclude the Solution

The time before the object hits the water is approximately 7.47 seconds.

Key Concepts

Gravitational Acceleration

Gravitational acceleration is a fundamental concept in physics that describes the acceleration of an object caused by the force of gravity acting on it. On Earth, this acceleration is approximately 9.81 meters per second squared ( 9.81 ext{ m/s}^2 ), and it dictates how quickly an object will speed up as it falls freely towards the ground.

It is important to understand that gravitational acceleration is a constant value, meaning that it does not change regardless of the object's mass. Thus, in a vacuum, where no air resistance is present, all objects fall at the same rate if dropped from the same height. This principle forms the basis of free fall physics, which is crucial for solving problems related to objects falling under the influence of gravity, like the exercise with the Hoover Dam Bridge.

• Acceleration due to Gravity: 9.81 ext{ m/s}^2 on Earth.
• Constant for All Objects: Independent of mass in a vacuum.
• Real-World Relevance: Crucial for predicting fall times.

Physics Problem Solving

Physics problem solving involves breaking down complicated scenarios into simpler, more understandable components. The key to solving any physics problem, such as the one involving the Hoover Dam Bridge, is to methodically extract the known information and apply the correct physical laws.

First, identify all the given variables and what you need to find. In this case, what we knew were the height of the bridge (274 m), the gravitational acceleration ( 9.81 ext{ m/s}^2 ), and the initial velocity (0 m/s). From there, we strategically chose the relevant kinematics equation to find the time it takes for the object to hit the water.

• Identify Given Variables: Know what is provided (height, acceleration, velocity).
• Choose the Right Equation: Pick a formula that links what you know to what you need.
• Solve Step by Step: Substitute known values, isolate variables, and solve.

By systematically approaching the solution, you can handle even complex problems with multiple variables. This process builds strong analytical skills, which are applicable beyond just physics.

Kinematics Equations

Kinematics equations describe the motion of objects and are used extensively in solving free-fall problems. These equations relate distance, velocity, time, and acceleration, allowing you to calculate one if the others are known.

In the problem about an object falling from the Hoover Dam Bridge, the kinematic equation used was:\[h = \frac{1}{2} g t^2\]This equation links the height fallen (h), gravitational acceleration (g), and the time (t). It comes from combining fundamental equations of motion to suit scenarios where the initial velocity is zero, like when an object is just dropped, and not thrown.

• Basic Motion Relationships: Connects distance, time, and acceleration.
• Special Case of Free Fall: Simplified version when initial velocity is zero.
• Derivation and Adaptation: Utilize different forms for various scenarios.

Understanding and applying kinematics equations empower you to solve a wide range of physics problems, making them an essential tool in physics education.