Problem 5 - Algebra 2

Question

The height \(h\) of an object \(t\) seconds after it is dropped is given by \(h=-\frac{1}{2} g t^{2}+h_{0},\) where \(h_{0}\) is the initial height and \(g\) is the acceleration due to gravity. The acceleration due to gravity near Earth's surface is \(9.8 \mathrm{m} / \mathrm{s}^{2},\) while on Jupiter it is 23.1 \(\mathrm{m} / \mathrm{s}^{2} .\) Suppose an object is dropped from an initial height of 100 meters from the surface of each planet. On which planet should the object reach the ground first?

Step-by-Step Solution

Answer

The object reaches the ground first on Jupiter.

Step 1: Write the Height Equation

The height equation for the object, when dropped, is given by \( h(t) = -\frac{1}{2} g t^2 + h_0 \), where \( g \) is the acceleration due to gravity, and \( h_0 \) is the initial height. We will use this equation to find out when the height becomes zero (the object hits the ground).

Step 2: Set the Equation to Zero

To find out when the object hits the ground, set \( h(t) = 0 \). The equation becomes: \[ -\frac{1}{2} g t^2 + h_0 = 0 \].

Step 3: Solve for Time on Earth

For Earth, with \( g = 9.8 \text{ m/s}^2 \) and \( h_0 = 100 \) meters, plug in the values: \[ 0 = -\frac{1}{2} (9.8) t^2 + 100 \]. Solve for \( t \): \[ t^2 = \frac{2 imes 100}{9.8} \] \[ t = \sqrt{20.41} \approx 4.52 \text{ seconds} \].

Step 4: Solve for Time on Jupiter

For Jupiter, with \( g = 23.1 \text{ m/s}^2 \), plug in the values: \[ 0 = -\frac{1}{2} (23.1) t^2 + 100 \]. Solve for \( t \): \[ t^2 = \frac{2 imes 100}{23.1} \] \[ t = \sqrt{8.66} \approx 2.94 \text{ seconds} \].

Step 5: Compare Times

The object takes approximately \( 4.52 \) seconds to hit the ground on Earth and approximately \( 2.94 \) seconds on Jupiter. Since \( 2.94 < 4.52 \), the object reaches the ground first on Jupiter.

Key Concepts

Acceleration due to Gravity

When you drop an object, it accelerates towards the ground due to a force called gravity. This acceleration, denoted as \( g \), is always directed downwards. On Earth, this acceleration is approximately \( 9.8 \, \text{m/s}^2 \). This means that every second, an object's speed increases by \( 9.8 \, \text{m/s} \).
Jupiter, being much larger than Earth, has a stronger gravitational pull. The acceleration due to gravity on Jupiter is \( 23.1 \, \text{m/s}^2 \).
This larger acceleration means objects on Jupiter accelerate faster when falling in comparison to Earth. This rapid change in speed contributes to them hitting the ground earlier. Understanding this concept helps us compare how objects fall on different planets.

Quadratic Equations

In the world of mathematics, quadratic equations play a crucial role in various calculations. These are equations where the highest exponent of a variable (usually \( t \) for time, in this case) is squared, which gives them the name 'quadratic'.
The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). In our problem, the equation to find when the object hits the ground simplifies into this form, where we need to find the time \( t \) when the height \( h \) equals zero.

• For Earth, the equation becomes \( -\frac{1}{2}(9.8) t^2 + 100 = 0 \).
• For Jupiter, it's \( -\frac{1}{2}(23.1) t^2 + 100 = 0 \).

Solve these equations by isolating \( t^2 \) on one side and taking the square root to find \( t \). This tells us the time taken for the object to reach the ground.

Planetary Comparison

Comparing planets involves understanding how different factors affect physical phenomena like gravity. Planets vary drastically in size, mass, and density, which influences their gravitational pull.
Earth's gravity is what we're most familiar with: objects fall with an acceleration of \( 9.8 \, \text{m/s}^2 \). Jupiter, being the largest planet in the solar system, has a stronger gravitational field, resulting in an acceleration of \( 23.1 \, \text{m/s}^2 \). This intense gravity means that an object dropped from the same height on both planets will hit the ground significantly faster on Jupiter.
This comparison is a practical demonstration of how planetary characteristics can define the motion of objects. Understanding these differences can be critical for space travel and understanding celestial mechanics in the universe.