Problem 102 - Elementary Algebra

Question

The height in feet of an object dropped from an airplane at 1,600 feet is given by \(h(t)=-16 t 2+1,600,\) where \(t\) is in seconds. a. How long will it take to reach half of the distance to the ground? b. How long will it take to travel the rest of the distance to the ground? Round off to the nearest hundredth of a second.

Step-by-Step Solution

Answer

It takes 7.07 seconds to reach halfway and an additional 2.93 seconds to hit the ground.

Step 1: Understanding the Problem

The problem provides us with a quadratic equation for the height of the object as a function of time, given by \( h(t) = -16t^2 + 1600 \). We need to find the time it takes for the object to reach half the distance to the ground and then the remaining time to reach the ground.

Step 2: Calculating Halfway Point

To find the time to reach halfway to the ground (800 feet), substitute \( h(t) = 800 \) into the equation: \(-16t^2 + 1600 = 800\).

Step 3: Solving for Time at Halfway

Set up the equation: \(-16t^2 + 1600 = 800\). Simplify to \(-16t^2 = -800\), then solve for \( t \) by dividing both sides by -16: \( t^2 = 50 \). Take the square root to find \( t = \sqrt{50} \approx 7.07 \) seconds.

Step 4: Calculating Time to Ground

To find the total time to reach the ground, set \( h(t) = 0 \). Solve \(-16t^2 + 1600 = 0\).

Step 5: Solving for Total Time to Ground

Set up the equation: \(-16t^2 + 1600 = 0\). Simplify to \(-16t^2 = -1600\), then divide by -16: \( t^2 = 100 \). Take the square root to find \( t = \sqrt{100} = 10 \) seconds.

Step 6: Calculating Remaining Time to Ground

Subtract the time to reach halfway (7.07 seconds) from the total time to reach the ground (10 seconds). This gives the additional time: \( 10 - 7.07 = 2.93 \) seconds.

Key Concepts

Gravity

Gravity is the force that pulls objects towards the center of the Earth. It gives us a straightforward way to model how objects fall when dropped. In physics, we use gravity to calculate how fast an object will move when it's falling. In our quadratic equation:

• The term \(-16t^2\) represents the acceleration caused by gravity.
• The '-16' is a constant derived from the gravitational acceleration near Earth's surface, equal to approximately \-32 \, \text{ft/s}^2\.

This is because the equation uses feet instead of meters. If it involved meters, the constant would be different. The object's initial height plays a big role in determining how long it will take to reach the ground. Here, the object starts from 1,600 feet up. So, the greater the height, the longer it will stay in the air.

Physics Applications

Physics applications of quadratic equations go beyond simple tools for calculating time and distance. These equations allow us to understand and predict natural events under the influence of various forces, especially gravity. For example, when you drop something from a height:

• The initial height is crucial for determining how long it will stay airborne.
• The structure of the equation helps separate the impact of distance and time.
• The \(-16t^2\) term in the height equation reflects the dynamic nature of falling objects subjected to gravity's constant pull.

Using these tools, engineers and scientists can design safer buildings by estimating how objects might fall during earthquakes or high wind conditions. This precise knowledge also aids in creating technologies such as parachutes, projecting how they will perform under gravity's force during descents.

Problem Solving

In tackling problems involving quadratic equations, a careful, step-by-step approach is essential. This means more than just plugging numbers into formulas. For question (a) in our exercise, we need to understand what 'halfway to the ground' truly means. It's not half the time, but half the distance, which is different.

• First, substitute into the quadratic equation to find the time when the height is 800 feet.
• Rearrange to isolate \(t^2\) for easy solving.
• Finally, take the square root, since \(t\) represents time and must be positive.

In question (b), we're finding the total time it takes to fall completely using the same principled approach before subtracting the halfway time. By breaking down these steps, especially in physics scenarios, we can enhance problem-solving skills and execute similar analyses in various fields.