Question

Use the following information. Scientists simulate a gravity-free environment called microgravity in free- fall situations. A similar microgravity environment can be felt on free-fall rides at amusement parks or when stepping off a high diving platform. The distance \(d\) (in meters) that an object that is dropped falls in \(t\) seconds can be modeled by the equation \(d=\frac{1}{2} g\left(t^{2}\right),\) where \(g\) is the acceleration due to gravity (9.8 meters per second per second). In Japan a 490 -meter-deep mine shaft has been converted into a microgravity facility. This creates the longest period of free fall currently available on Earth. How long will a period of free-fall be?

Short answer

The period of free fall will be approximately 30 seconds (rounded to the nearest second).

Step-by-step solution

Analyzing and rearranging the equation

The equation given is \(d = \frac{1}{2}gt^2\). We are asked to find \(t\) which is in the equation. The first step is to make \(t\) the subject in the above equation. Multiply both sides by 2 to eliminate the fraction which gives \(2d = gt^2\). Then, divide by \(g\) to solve for \(t^2\), so the equation becomes \(t^2 = \frac{2d}{g}\). Finally, we need to take the square root of both sides to solve for \(t\), and the equation becomes \(t = \sqrt{\frac{2d}{g}}\).

Substituting the values into the equation

Now we can substitute the given values into the rearranged equation. The distance \(d\) is 490 meters and we know that \(g\), acceleration due to gravity, is 9.8 meters per second per second. Hence, \(t = \sqrt{\frac{2*490}{9.8}}\).

Solving the equation

Solve the above equation using a calculator, and round the answer to the nearest second to find out how long the period of free fall will be.

Key concepts

Free-Fall Physics

Imagine dropping an object from a certain height and watching it accelerate towards the ground. This is an example of free-fall physics, a fundamental concept in classical mechanics. In a free-fall situation, the only force acting upon the object is gravity, which pulls it towards the center of the Earth.

In the absence of air resistance or any other force, all objects fall with the same acceleration, regardless of their mass. This means a feather and a hammer would hit the ground at the same time if dropped from the same height in a vacuum. This principle was famously demonstrated by the Apollo 15 mission on the Moon, where an astronaut conducted a live demonstration for the world to see.

Understanding the Equation

The equation of motion for an object in free fall is expressed as \(d = \frac{1}{2}gt^2\), where \(d\) represents the distance fallen, \(g\) is the acceleration due to gravity (9.8 m/s² on Earth), and \(t\) is the time in seconds. From this, we can understand that the distance an object falls is directly proportional to the square of the time it takes to fall. This quadratic dependence means that the distance traveled by a freely falling object increases rapidly as time progresses.

Microgravity

Microgravity, often mistaken as zero gravity, refers to the condition where the gravitational forces acting on an object are greatly reduced, simulating a near-weightless environment. It is not the complete absence of gravity, as the name might suggest, but rather a state where gravity's effects are much less noticeable.

Earth-bound facilities such as the Japanese mine shaft mentioned in the exercise create microgravity by allowing objects to fall freely, as they would in space, where the acceleration due to Earth's gravity (\(g\)) still exists, but its effects are not felt as strongly. Where astronauts aboard the International Space Station experience microgravity despite being in Earth's orbit, free-falling objects in such facilities briefly encounter similar conditions during their descent.

Simulating Space on Earth

Simulating such conditions is crucial for experiments that require a gravity-free environment, such as testing equipment for space missions or researching biological processes that may behave differently in low-gravity surroundings. These brief windows of microgravity can be incredibly valuable, allowing scientists to study phenomena for longer periods than the few seconds available during parabolic flights, which are typically used for shorter duration microgravity experiments.

Acceleration due to Gravity

Acceleration due to gravity is a constant value representing the rate at which the speed of an object increases as it falls freely toward Earth. On our planet's surface, this value is approximately 9.8 m/s², meaning that for every second an object is in free fall, its velocity increases by 9.8 meters per second.

This acceleration is a crucial component in the equation \(d = \frac{1}{2}gt^2\), shaping our understanding of how objects move under the influence of Earth's gravitational force. Regardless of where you are on the planet, the acceleration due to gravity remains fairly consistent, although there are minor variations due to altitude and the Earth's shape.

Finding the Free-Fall Time

Using this principle, the time it takes for an object to fall a certain distance can be calculated. In the case of the Japanese mine shaft, we rearrange the motion equation to solve for \(t\), as seen in the exercise. By substituting the known value of \(g\) and the distance \(d = 490 \text{ meters}\), we find the time the object will be in free fall, thus experiencing microgravity: \(t = \sqrt{\frac{2d}{g}}\). This calculation demonstrates how acceleration due to gravity is instrumental in predictive equations related to the movement of objects on Earth.