Problem 75 - College Algebra and Calculus: An Applied Approach

Question

Assume that air resistance is negligible, which implies that the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) is a reasonable model Wind Resistance At the same time a skydiver jumps from an airplane 13,000 feet above the ground, a steel ball is dropped from the plane. Because of air resistance, it takes the skydiver 67 seconds to freefall to a height of 3000 feet where the parachute opens. The steel ball has relatively no air resistance, so its height can be modeled by the position equation. How much faster does the ball reach a height of 3000 feet than the skydiver?

Step-by-Step Solution

Answer

The steel ball reaches a height of 3000 feet 38 seconds faster than the skydiver.

Step 1: Understanding the Initial Conditions

For the steel ball, physical conditions are as follows: since it is dropped and not thrown, its initial velocity \(v_{0}\) is 0. The initial height \(s_{0}\) is 13,000 feet (the height from which it is dropped), and we're finding when it falls to a height \(s\) of 3000 feet. Thus we can replace these values in the position equation.

Step 2: Setting up the Equation

By replacing the known values into the position equation, we get: \(3000=-16 t^{2}+0 \cdot t+ 13000\). Note that since \(v_{0}=0\), the term involving it will become zero and we are left with: 3000=-16\(t^{2}\) + 13000. This equation can be rearranged to isolate \(t^{2}\). Yielding \(t^{2} = \frac {13000-3000}{16}\).

Step 3: Calculating the Time

Solving that equation gives us the time squared, so we must take the square root to find the time \(t\). The square root part is important since we're looking for a time, not a time squared. So, \(t = \sqrt\frac {13000-3000}{16}\), which simplifies further to \(t \approx 29\) seconds.

Step 4: Comparing times

Lastly, subtract the time it takes for the steel ball to fall to 3000 feet from the time it takes the skydiver. This gives \(67 - 29 = 38\). Thus, the steel ball reaches a height of 3000 feet 38 seconds faster than the skydiver.

Key Concepts

Free Fall

Free fall is a type of motion where an object is moving only under the influence of gravity. This means that the object accelerates downwards at a consistent rate due to gravity. In most cases on Earth, this acceleration is approximately

• 9.8 m/s².
• However, when using other units, such as feet, this value is generally rounded to 32 ft/s².

In many physics problems, including the skydiver and steel ball exercise, it is assumed that air resistance is negligible, allowing us to use the simplified model of free fall. This means that the only force acting on the object is gravity, completely ignoring any frictional force from air resistance that would normally slow it down.
Remember, in a true free fall scenario, all objects fall at the same rate regardless of their mass, due to gravity acting equally on everything.

Position Equation

The position equation is a fundamental formula used to determine the position of an object in free fall at any given time. In the form \(s = -16t^2 + v_0 t + s_0\),

• \(s\) represents the position of the object at time \(t\),
• \(v_0\) is the initial velocity, and
• \(s_0\) is the initial height.

This equation is derived from the kinematic equations of motion. It allows you to determine the height of an object at any point in its descent, assuming vertical motion due to gravity without any other forces acting upon it (like air resistance). By substituting known values into this equation, you can solve for unknowns, such as the time it takes for an object to fall to a particular height.

Initial Conditions

Initial conditions in motion problems are vital for setting the stage for accurate calculations. In our exercise, the initial conditions involve the starting parameters of the motion:

• The steel ball starts from a stationary position, so its initial velocity \(v_0\) is 0 ft/s.
• It drops from a height of 13,000 feet, making \(s_0\) = 13,000 feet.

These conditions allow us to simplify the position equation by eliminating the initial velocity component, leading to a clear calculation path involving only the time and initial height to find a specific position at any point during the free fall. Getting the initial conditions right can help solve what might seem like a complex problem with straightforward algebraic manipulations.

Air Resistance

Air resistance, or drag, is the force that opposes the motion of an object through the air. It's particularly important to consider when high speeds are involved or when the object has a large surface area. In the context of our exercise, however, air resistance for the steel ball is negligible. This assumption simplifies the problem considerably since we can then analyze the motion using the free-fall equations without adding the complexities that air resistance introduces.

Why Air Resistance Matters

In real-world scenarios, air resistance becomes a significant factor, especially for objects that move fast or have a large surface area relative to their mass, like a skydiver. It acts to slow down these objects, reducing the acceleration they experience due to gravity alone. For precise calculations in such situations, air resistance must be measured or estimated and factored into the equations of motion, often leading to a much more complex set of equations. By assuming no air resistance, often an adequate first step in simpler physics problems, calculations are simplified, focusing purely on the gravitational force at play.