Problem 7 - 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. Do the times to reach the ground seem reasonable? Explain.

Step-by-Step Solution

Answer

On Earth, the time is approximately 4.52 seconds; on Jupiter, it's about 2.94 seconds. The times are reasonable given the difference in gravity.

Step 1: Formulate the Problem for Earth

We start by considering the equation for the height of an object in free fall, given by \( h = -\frac{1}{2} g t^{2} + h_{0} \). For Earth, we have \( g = 9.8 \, \text{m/s}^2 \) and \( h_0 = 100 \, \text{meters} \). Our goal is to find the time \( t \) when \( h = 0 \), indicating the object has reached the ground.

Step 2: Solve for Time on Earth

Set \( h = 0 \) in the height equation:\[ 0 = -\frac{1}{2} (9.8) t^2 + 100 \]Simplifying, we get:\[ -4.9t^2 + 100 = 0 \]Rearranging gives:\[ 4.9t^2 = 100 \]Divide both sides by 4.9:\[ t^2 = \frac{100}{4.9} \]Solve for \( t \):\[ t = \sqrt{\frac{100}{4.9}} \approx 4.52 \, \text{seconds} \]

Step 3: Formulate the Problem for Jupiter

Now consider the same scenario on Jupiter where \( g = 23.1 \, \text{m/s}^2 \) and \( h_0 = 100 \, \text{meters} \). Again, find the time \( t \) when the object reaches the ground (\( h = 0 \)).

Step 4: Solve for Time on Jupiter

Set \( h = 0 \) in the height equation:\[ 0 = -\frac{1}{2} (23.1) t^2 + 100 \]Simplifying, we get:\[ -11.55t^2 + 100 = 0 \]Rearranging gives:\[ 11.55t^2 = 100 \]Divide both sides by 11.55:\[ t^2 = \frac{100}{11.55} \]Solve for \( t \):\[ t = \sqrt{\frac{100}{11.55}} \approx 2.94 \, \text{seconds} \]

Step 5: Compare Times and Assess Reasonableness

Comparing the times calculated, on Earth the object takes approximately 4.52 seconds, while on Jupiter it takes about 2.94 seconds to reach the ground. This difference is expected because the gravitational acceleration is greater on Jupiter, causing the object to fall faster compared to Earth.

Key Concepts

Acceleration due to Gravity

The acceleration due to gravity, often denoted by the symbol \( g \), is a fundamental concept in physics. It describes how fast an object accelerates when falling freely close to a planetary surface. On Earth, the acceleration due to gravity is \( 9.8 \, \text{m/s}^2 \). This means that for every second an object is in free fall, its velocity increases by approximately \( 9.8 \, \text{m/s} \). On Jupiter, however, gravity is much stronger, with \( g = 23.1 \, \text{m/s}^2 \).

Gravity is the cause of an object's free fall, and the stronger it is, the faster an object will accelerate. This explains why an object dropped on Jupiter reaches the ground more quickly compared to one dropped on Earth. The magnitude of gravity affects the trajectory and speed of an object in free fall significantly.

Quadratic Equations

The equation for calculating the time it takes for an object to hit the ground when dropped from a specific height involves solving a quadratic equation. In the context of free fall, the height equation is expressed as \( h = -\frac{1}{2} g t^2 + h_0 \). This equation is derived from the kinematic equations of motion.

To find the time \( t \), we set the height \( h \) to zero, signifying the moment the object hits the ground. The equation simplifies into a standard form of a quadratic equation, \( ax^2 + bx + c = 0 \). In this problem, it translates to:

• \( a = -\frac{1}{2} g \)
• \( b = 0 \)
• \( c = h_0 \)

The task then is to solve for \( t \), using methods such as factoring, completing the square, or applying the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Understanding how to manipulate and solve these equations is essential in physics.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces causing the motion. When analyzing free fall motion, we use kinematic equations to describe the position, velocity, and acceleration of the falling object over time.

The key kinematic equation used here is: \[ h = -\frac{1}{2} g t^2 + h_0 \] which helps us determine the time at which the object reaches the ground. Kinematics allows us to predict how long it will take for an object to fall to the surface when released from a known height. It takes into account initial velocity, which in free fall typically is zero. Understanding kinematics is critical to solving problems involving any object's motion.

Planetary Motion

The study of planetary motion delves into how different celestial bodies interact through gravitational forces. It explains why planets with stronger gravity, such as Jupiter, affect free-falling objects differently than those with weaker gravity, such as Earth.

1. **Gravitational Influence**: Each planet's gravitational pull depends on its mass and size. A more massive planet like Jupiter exerts a stronger force per unit of mass. 2. **Falling Objects Behavior**: On larger planets, objects tend to fall faster due to increased gravitational force. This phenomenon is evident in the shorter fall time for objects on Jupiter compared to Earth. 3. **Applications in Physics and Space Exploration**: Understanding planetary motion and differing gravitational forces helps scientists predict satellite behavior and plan space missions effectively. Planetary motion provides a broader understanding of how gravity varies across the solar system and influences objects and their motion.