Free fall with air resistance An object is dropped straight...

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 $5 \mathrm{sec},$ as shown: $$\begin{array}{cccccc}{t} & {0} & {1} & {2} & {3} & {4} & {5} \\ \hline a & {32.00} & {19.41} & {11.77} & {7.14} & {4.33} & {2.63}\end{array}$$ 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 $t=3$ .

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 $5 \mathrm{sec},$ as shown:
$$\begin{array}{cccccc}{t} & {0} & {1} & {2} & {3} & {4} & {5} \\ \hline a & {32.00} & {19.41} & {11.77} & {7.14} & {4.33} & {2.63}\end{array}$$
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 $t=3$ .

Key Concepts

Free Fall Dynamics with Air Resistance

This concept involves the motion of objects under the influence of gravity while also being opposed by air resistance. Air resistance acts as a drag force that increases with the object's speed, causing the acceleration to decrease over time even though the object is still speeding up. This interplay between gravitational force and air resistance is key to understanding why objects do not accelerate indefinitely in a real-world free fall scenario.

Acceleration as a Rate of Change of Velocity

Acceleration is defined as the rate at which an object's velocity changes with time. In kinematics, acceleration is the second derivative of position with respect to time. This concept is critical when analyzing free fall because it allows us to understand how quickly the velocity of a falling object increases, even when that increase is moderated by factors like air resistance.

Velocity as the Integral of Acceleration

Velocity is obtained by integrating the acceleration function over a given time interval. In the context of free fall with air resistance, where acceleration is not constant, the speed of the object at a particular time can be estimated by summing (integrating) the acceleration values over time. This process of integration is key to transitioning from knowing acceleration to determining the velocity or speed of the object.

Numerical Integration and Riemann Sums

Numerical integration techniques, such as Riemann sums, are used to approximate the value of definite integrals when the function is known only at certain points. This is particularly useful when dealing with real-world experimental data, where acceleration is recorded at discrete time intervals. By partitioning the time into intervals and summing up the contributions of each, one can approximate the overall change in velocity and, subsequently, the distance fallen.

Upper and Lower Riemann Sum Approximations

When applying Riemann sums, the choice between using left-endpoint or right-endpoint evaluations leads to overestimates or underestimates, respectively. For a decreasing function, a left Riemann sum typically provides an upper estimate while a right Riemann sum gives a lower estimate. This concept is fundamental in estimating quantities like speed and distance when the underlying function (acceleration or velocity) changes over time.

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