Problem 13 - University Calculus: Early Transcendentals

Question

Free fall with air resistance An object is dropped straight down from a helicopter. The object falls faster and faster but its acceleration (rate of change of its velocity) decreases over time because of air resistance. The acceleration is measured in \(\mathrm{ft} / \mathrm{sec}^{2}\) and recorded every second after the drop for 5 sec, as shown: $$\begin{array}{c|cccccc} t & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline a & 32.00 & 19.41 & 11.77 & 7.14 & 4.33 & 2.63 \end{array}$$ a. Find an upper estimate for the speed when \(t=5.\) b. Find a lower estimate for the speed when \(t=5.\) c. Find an upper estimate for the distance fallen when \(t=3.\)

Step-by-Step Solution

Answer

a. 74.65 ft/s, b. 45.28 ft/s, c. 63.18 ft.

Step 1: Understanding Upper and Lower Estimates for Speed

To find the speed of an object falling with varying acceleration, we need to calculate the integral of the acceleration over time. We can use approximation techniques like the trapezoidal or midpoint rule to estimate the velocity when we do not know a continuous function.

Step 2: Upper Estimate for Speed at t=5

For an upper estimate of velocity at \( t = 5 \), we can use the left Riemann sum, where we take the initial five acceleration values up to \( t = 5 \). Each time interval is 1 second, so compute:\[v_5^{upper} = \Delta t \times (a_0 + a_1 + a_2 + a_3 + a_4) = 1 \times (32.00 + 19.41 + 11.77 + 7.14 + 4.33)\]Calculating the sum gives:\[ v_5^{upper} = 74.65 \text{ ft/s} \]

Step 3: Lower Estimate for Speed at t=5

For a lower estimate of velocity at \( t = 5 \), we use the right Riemann sum, where we consider the last five acceleration values from \( t = 1 \) to \( t = 5 \). Calculate:\[v_5^{lower} = \Delta t \times (a_1 + a_2 + a_3 + a_4 + a_5) = 1 \times (19.41 + 11.77 + 7.14 + 4.33 + 2.63)\]Calculating the sum gives:\[ v_5^{lower} = 45.28 \text{ ft/s} \]

Step 4: Upper Estimate for Distance Fallen at t=3

To estimate the distance fallen by \( t = 3 \), calculate the upper sum of velocity from \( t = 0 \) to \( t = 3 \). Since we don't have velocity directly, we will integrate acceleration over time using the left Riemann sum to estimate displacement:\[s_3^{upper} = \Delta t \times (a_0 + a_1 + a_2) = 1 \times (32.00 + 19.41 + 11.77)\]Calculating the sum gives:\[ s_3^{upper} = 63.18 \text{ ft} \]

Key Concepts

Integral Calculus

Integral calculus involves calculating the area under a curve. When you integrate a function, you're finding the total "accumulation" of a quantity. This is crucial when dealing with rates, like acceleration, because you can find velocities or distances by integrating these rates.
To find velocity from acceleration, you integrate the acceleration function over time. If a function describing the rates continues over an interval, integrals

• Sum infinitesimal quantities for exact values.
• Use approximation methods, like trapezoidal or midpoint rules, for functions without specific equations.

Understanding integral calculus is essential for solving problems involving real-life applications, like calculating velocity and distance for falling objects.

Riemann Sums

Riemann sums are a technique to approximate integrals. When you can't integrate precisely, Riemann sums break down the curve into rectangles. The area of each rectangle gives an approximation of the integral’s value. This method is especially useful in situations where data is given at discrete intervals, not a continuous function.
For estimating velocity of a falling object:

• The **left Riemann sum** uses values from the start of intervals, giving an upper estimate because it covers the largest initial observation period.
• The **right Riemann sum** considers values from the end, providing a lower estimate as it uses smaller or declining values.

This makes Riemann sums a practical approach to approximating speed or distance without direct integrating functions.

Velocity Estimation

Velocity estimation becomes necessary when acceleration changes with time. While the process assumes constant acceleration sometimes, real-world forces change this. Calculating velocity from changing acceleration requires integrating or using methods like Riemann sums.
In the given exercise, velocity is estimated over specific intervals using the sum of discrete acceleration values and multiplied by time to calculate speed:
- **Upper estimate**: Uses left Riemann sum, assuming maximum speed by initially higher accelerations. - **Lower estimate**: Uses right Riemann sum, considering reduced speed due to decreasing acceleration.
This way, we approximate velocities efficiently, crucial in situations where detailed function models are absent.

Air Resistance Effect

Air resistance affects how objects fall by reducing acceleration over time. In a vacuum, objects accelerate constantly due to gravity. However, when air is present, this resistance acts to decelerate, producing gradually decreasing acceleration values as observed in the exercise.
Air resistance is crucial to consider because:

• It influences terminal velocity, the point where acceleration stops and speed remains constant.
• Affects calculations significantly in physics problems, adjusting realistic estimations of speed and distance.
• Requires understanding beyond basic mechanics, often involving more complex calculus to model accurately.

Thus, including air resistance in calculations is essential for determining true motion characteristics of falling bodies with accuracy.