Problem 116 - Fundamentals of Physics

Question

Most important in an investigation of an airplane crash by the U.S. National Transportation Safety Board is the data stored on the airplane's flight-data recorder, commonly called the "black box" in spite of its orange coloring and reflective tape. The recorder is engineered to withstand a crash with an average deceleration of magnitude \(3400 \mathrm{~g}\) during a time interval of \(6.50 \mathrm{~ms}\). In such a crash, if the recorder and airplane have zero speed at the end of that time interval, what is their speed at the beginning of the interval?

Step-by-Step Solution

Answer

The initial speed is approximately 216.86 m/s.

Step 1: Understand the Given Data

We are given the average deceleration of magnitude \(3400\, \text{g}\) and the time interval of deceleration as \(6.50\, \text{ms}\). The recorder reaches zero speed at the end of this time interval. We need to calculate the initial speed at the beginning of the interval.

Step 2: Convert Gravity Units and Time

First, convert the given deceleration from gravity units (\(\text{g}\)) to meters per second squared (\(\text{m/s}^2\)). Since \(1\, \text{g} = 9.8\, \text{m/s}^2\), the deceleration is \(3400 \times 9.8\, \text{m/s}^2\). Convert time from milliseconds to seconds: \(6.50\, \text{ms} = 6.50 \times 10^{-3}\, \text{s}\).

Step 3: Apply the Deceleration Formula

Using the formula for uniform acceleration (deceleration in this case), \(v_f = v_i + a \cdot t\), where \(v_f\) is the final speed (0 in this case), \(v_i\) is the initial speed, \(a\) is the acceleration (negative for deceleration), and \(t\) is time, we solve for \(v_i\):\[0 = v_i - 3400 \times 9.8 \times 6.50 \times 10^{-3}\]

Step 4: Solve for Initial Speed

Rearrange the formula to solve for \(v_i\):\[v_i = 3400 \times 9.8 \times 6.50 \times 10^{-3}\] Calculate \(v_i\).

Step 5: Calculate the Numerical Value

Perform the calculation:\[v_i = 3400 \times 9.8 \times 6.50 \times 10^{-3} = 216.86\, \text{m/s}\]

Key Concepts

Deceleration Calculation

Deceleration is simply negative acceleration. It represents slowing down. In physics problems like this, we often need to know how much an object slows over time. Calculations involving deceleration allow us to figure out what an object’s speed was before slowing down to a stop. To calculate deceleration, use the formula: \[v_f = v_i + a \cdot t\]- **Where:** - \(v_f\) is the final velocity (0 if it comes to a stop) - \(v_i\) is the initial velocity - \(a\) is acceleration (negative for deceleration) - \(t\) is timeIn our problem, the plane’s black box recorder is decelerating from an initial velocity \(v_i\) to a final velocity \(0\). The recorder withstands a deceleration of magnitude \(3400 \text{ g’s}\). With the total deceleration time then given at \(6.50 \text{ ms}\), we figure out the initial speed needed to deliver to the recorder so it comes to a stop at zero speed in that time.

Flight Data Recorder

The flight data recorder is crucial for investigating airplane crashes. Often called the 'black box,' though it is actually orange for visibility, it records critical flight information like speed, altitude, and flight data during flight. Its ability to withstand crashes due to substantial forces ensures it preserves key flight information. In our physics problem, understanding the deceleration involved that the black box can endure gives insight into how airplanes manage to safely store vital data even under extreme conditions like a crash. This information is central to investigations as it aids in determining the series of events leading up to a crash.

Unit Conversion in Physics

Unit conversion is a frequent requirement in physics problems. Converting between units ensures that all terms in an equation are compatible and calculations are correct. For instance, in the problem above, we need to convert units of acceleration and time.- **Gravity Units to \(\text{m/s}^2\):** - To convert \(3400\,\text{g}\) to \(\text{m/s}^2\), multiply by \(9.8\,\text{m/s}^2\) since \(1\,\text{g} = 9.8\,\text{m/s}^2\). - **Time from Milliseconds to Seconds:** - Convert \(6.50\,\text{ms}\) to seconds by multiplying by \(10^{-3}\).Using the proper units ensures mathematical and physical consistency in computations, allowing us to solve complex problems accurately. By converting these units, we can effectively apply equations of motion and complete calculations like finding initial speeds during deceleration.