Problem 99 - Algebra 1

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). If you want to double the free-fall time, how much do you have to increase the height from which the object was dropped?

Step-by-Step Solution

Answer

To double the free-fall time, the height from which the object was dropped must be quadrupled.

Step 1: Understanding the problem

The equation \(d=\frac{1}{2} g\left(t^{2}\right)\) describes how the distance d fallen for a duration t in a constant gravity environment. Here, g is the acceleration due to gravity and t is the time in seconds. The problem asks what happens to distance \(d\) when time \(t\) is doubled.

Step 2: Doubling the time

First, let's keep the initial time as \(t\) and the initial distance as \(d\). When the time, \(t\), is doubled, the new time becomes \(2t\). We substitute \(2t\) in place of \(t\) in the equation. It becomes \(d' = \frac{1}{2} g \left(2t\right)^2\).

Step 3: Simplifying the equation

Simplifying the above equation gives \(d' = \frac{1}{2} g \cdot 4t^2 = 2gt^2\). Here \(d'\) is the new distance when the time is doubled.

Step 4: Comparing the initial and new distances

The initial distance is \(d = \frac{1}{2}gt^2\) and the new distance when time is doubled is \(d' = 2gt^2\). Thus, when the time is doubled, the distance covered is four times the original distance.

Key Concepts

Acceleration due to Gravity

The acceleration due to gravity, often denoted as \( g \), is pivotal in understanding the dynamics of free-fall. Imagine dropping an object from a certain height. Due to earth's gravitational pull, the object will accelerate at approximately 9.8 meters per second squared. This constant acceleration means each second, the object's velocity increases by 9.8 meters per second.

In the absence of resistance, such as air or other forces, this consistent acceleration helps us predict motion during free-fall.

• This makes \( g \) a key parameter in physics problems, especially those involving vertical motion.
• The value 9.8 m/s² informs how an object's velocity and distance change over time.
• Understanding \( g \) is crucial for environments simulating microgravity, like in amusement park rides or space stations.

By applying this knowledge, you can predict how long it will take for an object to hit the ground when dropped, or inversely, how high it must fall from to reach a specific speed.

Quadratic Equation

The equation used to model free-fall distance \(d = \frac{1}{2} g(t^2)\) is a quadratic equation. This type of equation is characterized by the variable being squared, making it essential to understand how time \(t\) plays a role in determining distance.

Here’s what happens when you double the time:\(\)

• Substituting \(2t\) for \(t\) in the equation \(d = \frac{1}{2}g(t^2)\) results in \(d' = \frac{1}{2}g(2t)^2\).
• Expanding \((2t)^2\) gives \(4t^2\), leading to \(d' = 2gt^2\). This shows that with the time doubled, the fall distance becomes four times greater.

Quadratic equations are powerful in physics because they describe scenarios where a rate of change (like velocity in free-fall) itself changes consistently over time, reflecting how non-linear relationships behave in real-world applications.

Physics of Motion

The physics of motion encompasses all the principles that describe how objects move. In free-fall scenarios, motion is dictated by gravity alone, setting aside friction or any other forces. Utilizing equations derived from Newton's laws, physicists can accurately predict how an object behaves when in motion. With \(d = \frac{1}{2}g(t^2)\), we see how crucial each variable is.

• Time \(t\) is squared, signifying that distance doesn’t increase linearly as time progresses. Instead, it accelerates, as evident from the impact of doubling \(t\) on distance \(d\).
• Understanding motion’s principles is not only essential in laboratories or physics classrooms but is practical in designing rides, calculating space mission trajectories, or even everyday tasks like throwing a ball.

By mastering the basics of free-fall and equations governing motion, students can bridge the gap between theoretical physics and practical applications, enhancing their ability to solve complex real-world problems.