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 \(5 \mathrm{sec},\) as shown in the table below. $$\begin{array}{c|ccccc}{t} & {0} & {1} & {2} & {3} & {4} & {5} \\ \hline a & {32.00} & {19.41} & {11.77} & {7.14} & {4.33} & {2.63}\end{array}$$ (a) Use \(L R A M_{5}\) to find an upper estimate for the speed when \(t\)=5. (b) Use RRAM \(_{5}\) to find a lower estimate for the speed when \(t=5 .\) (c) Use upper estimates for the speed during the first second, second second, and third second to find an upper estimate for the distance fallen when \(t=3 .\)

Short answer

Upper speed estimate (LRAM) at \(t=5\) sec is \(74.65\) ft/sec. Lower speed estimate (RRAM) at \(t=5\) sec is \(45.28\) ft/sec. The upper estimate for distance fallen when \(t=3\) sec is \(51.41\) ft.

Step-by-step solution

Compute Upper Estimate (LRAM)

To compute LRAM, consider that the acceleration value at the start of each interval (or each second) is the rate of change of speed for that interval. That is, to find the speed at \(t=5\) sec, multiply the acceleration at each time \(t\) by the time interval and sum the results. Using the given table, the estimated speed at \(t=5\) sec will be \((32.0 * 1) + (19.41 * 1) + (11.77 * 1) + (7.14 * 1) + (4.33 * 1) = 74.65\) ft/sec.

Compute Lower Estimate (RRAM)

To compute RRAM, consider that the acceleration value at the end of each time interval (or each second) is the rate of change of speed for that interval. Thus, to find the speed at \(t=5\) sec, multiply the acceleration at each time \(t+1\) by the time interval and sum the results. However, this method does not consider the acceleration at \(t=0\). So, using the given table, the estimated speed at \(t=5\) sec will be \((19.41 * 1) + (11.77 * 1) + (7.14 * 1) + (4.33 * 1) + (2.63 * 1) = 45.28\) ft/sec.

Calculate Upper Estimate for Distance fallen in first 3 seconds

To find the upper estimate for the distance fallen in the first three seconds, consider the calculated speeds at the end of each second as the rate of displacement for the next second. For the first second, the object is still stationary, so displacement is zero. Then, for the second and third seconds, displacement is equal to speed at the end of these seconds, which can be calculated using LRAM from the acceleration data. Hence, the estimated distances at \(t=1, 2, 3\) sec are \((0 * 1) + (32.0 * 1) + (19.41 * 1) = 51.41\) ft.

Key concepts

LRAM and RRAM approximation

Imagine you want to estimate the total amount of water collected in several buckets during different periods of rain, but you only have measurements at the start or end of each period. This is analogous to what we do with Left Riemann Sums (LRAM) and Right Riemann Sums (RRAM) when estimating physical quantities like speed and distance.

In the context of our free-fall problem with air resistance, LRAM uses the acceleration value at the beginning of each time interval to approximate speed. Think of it like using the amount of rain at the beginning of each period to estimate the total rain collected—it's a conservative guess.
On the other hand, RRAM uses the acceleration at the end of each time interval for approximation, like assuming the rain at the end of the period continued throughout, which might underestimate if the rain was tapering off. In essence, LRAM and RRAM provide us with upper and lower bounds, respectively, for the speed of our falling object over time.

These methods are part of the fundamental concepts of calculus and are used to estimate integrals when exact calculations are difficult or impossible. They're especially handy when dealing with real-world data that's recorded at specific intervals, just like our acceleration due to gravity measurements.

Acceleration due to gravity

Acceleration, or the rate at which velocity changes, is a core concept when dealing with motion. Particularly, the acceleration due to gravity, usually represented by the symbol \(g\), is approximately \(9.8 \frac{m}{s^2}\) or \(32 \frac{ft}{s^2}\) near the surface of the Earth.

This figure represents the speed gained per second by an object in free fall, assuming no other forces act on it—particularly air resistance, which in reality does slow down falling objects like our helicopter-dropped example. The varying acceleration values in our problem highlight the impact of air resistance, showing that as the object falls faster, the resistance pushes back harder, and thus the acceleration decreases over time. Calculating these changes in acceleration is crucial for predicting the object's eventual speed and distance traveled, which is precisely what our textbook problem involves.

Estimating speed and distance

In physics, estimating speed and distance in dynamic situations like free fall can be challenging due to factors like air resistance. Our example problem simplifies this complex scenario by providing acceleration data at one-second intervals, allowing us to use approximation methods.

To estimate speed, you multiply the acceleration at each time by the time interval, summing up the results. Think of it as if you're calculating the object's growing speed bank account, with each second's acceleration contributing to the total. For distance estimation, you'd tally up these speed 'deposits' for each second, getting a sense of how far the object has traveled.
However, remember that due to air resistance, acceleration is not constant, making our RRAM (lower estimate) and LRAM (upper estimate) aproaches necessary to bound our predictions. This process is crucial in many fields, from aerospace to sports, where precision in predicting motion can make big differences in performance and safety.

Calculus in physics

Calculus is the mathematical toolkit for understanding change—perfect for analyzing motion in physics. The key tools from calculus used in our textbook problem are derivatives and integrals. Derivatives describe the rate of change, like velocity and acceleration.

The integral, which LRAM and RRAM approximate, essentially adds up these tiny changes to give an overall effect, like total distance traveled. It's a pivot from focusing on individual instants to evaluating the cumulative effect over time.
Imagine watching a movie frame by frame (derivatives) and then watching the whole movie to see the entire story (integral). Using calculus, physicists can model real-life scenarios, such as our falling object, predicting its behavior by piecing together information about its changing velocity and location over time.