Problem 88 - Algebra and Trigonometry

Question

Suppose an object is dropped from a height \(h_{0}\) above the ground. Then its height after \(t\) seconds is given by \(h=-16 t^{2}+h_{0},\) where \(h\) is measured in feet. Use this information to solve the problem. A ball is dropped from the top of a building 96 \(\mathrm{ft}\) tall. (a) How long will it take to fall half the distance to ground level? (b) How long will it take to fall to ground level?

Step-by-Step Solution

Answer

(a) Approximately 1.73 seconds. (b) Approximately 2.45 seconds.

Step 1: Understanding the problem

The problem provides us with the formula for height after a certain time: \[ h = -16t^2 + h_0 \]where \( h_0 \) is the initial height and \( t \) is the time in seconds.We need to find the time when the height becomes half of the initial height \( h_0 \) and when it hits the ground (height becomes 0).

Step 2: Identifying initial condition

Given the initial height \( h_0 = 96 \text{ ft} \). This information will help us determine the time when the ball is at half-height and at ground level.

Step 3: Half distance to ground

Half of the initial height is \( \frac{h_0}{2} = \frac{96}{2} = 48 \text{ ft} \).Substitute \( h = 48 \) into the height formula:\[ 48 = -16t^2 + 96 \]Rearrange to find \( t \):\[ -16t^2 = 48 - 96 \]\[ -16t^2 = -48 \]Solve for \( t^2 \):\[ t^2 = \frac{-48}{-16} = 3 \]\[ t = \sqrt{3} \approx 1.73 \text{ seconds} \]

Step 4: Falling to the ground

The ball reaches the ground when the height \( h = 0 \).Substitute \( h = 0 \) into the formula:\[ 0 = -16t^2 + 96 \]Rearrange to solve for \( t^2 \):\[ -16t^2 = -96 \]\[ t^2 = \frac{-96}{-16} = 6 \]\[ t = \sqrt{6} \approx 2.45 \text{ seconds} \]

Key Concepts

Physics of Free Fall

The concept of free fall in physics describes the motion of an object that is only influenced by gravity, with no other forces acting upon it. This is a fundamental principle when studying motion in physics, especially for objects in a gravitational field.

• When an object is in free fall, it accelerates downward at a constant rate due to gravity.
• The lack of air resistance or other forces allows it to have a predictable, uniform motion characterized by constant acceleration.

In the case of the exercise, the ball is dropped from a height without any initial velocity, which means it starts falling due to the force of gravity alone. This scenario allows the problem to be simplified using physics equations for free fall, such as the given quadratic equation for height over time. Without considering air resistance, the only acceleration here is due to gravity, which simplifies the calculation of the time taken for the ball to fall to different heights.

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Solving them can determine either time, distance, or another variable depending on the setup.

• These equations often have two solutions, one for when the value of the variable is increasing and one when it is decreasing.
• In physics problems like the one given, solving a quadratic equation helps find the time it takes for an object to reach a particular height when falling freely.

For this exercise, the equation for the fallen height \( h = -16t^2 + h_0 \) becomes our focus. Here, 96 feet being the starting height, allows the equation to be set to find the times when the ball is at half its height and when it reaches ground level.
Requiring simple rearrangement and solving processes, like taking the square root of the outcome, these steps are crucial in predicting motion in gravitational fields.

Gravity in Feet Per Second Squared

Gravity is a universal force that causes everything to fall towards the Earth at a constant acceleration. In most calculations involving free fall near Earth's surface, gravity is measured in units familiar to the local measurement system.

• In the case of feet per second squared, the constants used symbolize the acceleration due to gravity as \( 32 \text{ ft/s}^2 \); however, in equations, \(-16\) is used because height decreases as time increases.
• This conversion and representation facilitate the calculation of free fall in environments using the customary system.

The problem uses this measurement to explain how the speed of the falling object changes over time. This constant acceleration is essential in determining how long it takes the object to fall from one height to another.
The use of gravity in these terms is crucial for directly applying the associated laws of motion, simplifying calculations needed for diverse physics problems in teaching environments.