Question

A Boeing 737 aircraft has a mass of \(150,000 \mathrm{~kg}\), and a cruising velocity of \(720 \mathrm{~km} / \mathrm{hr}\). Its engines can create a total thrust of \(200,000 \mathrm{~N}\). If air resistance, change in altitude, and fuel consumption can be ignored, how long does it take for the plane to reach its cruising velocity starting from rest? 1\. \(100 \mathrm{~s}\) 2\. \(150 \mathrm{~s}\) 3\. \(540 \mathrm{~s}\) 4\. \(1944 \mathrm{~s}\)

Short answer

2. $150 \mathrm{~s}$

Step-by-step solution

Determine the acceleration of the aircraft

To find the acceleration (a), we will use the formula \(F = ma\), where F is the total thrust (200,000 N) and m is the mass of the aircraft (150,000 kg). Rearranging the formula to solve for a, we get: \[a = \frac{F}{m}\] Substituting the given values, we get: \[a = \frac{200,000 \mathrm{~N}}{150,000 \mathrm{~kg}}\] Simplifying the equation, we find that the acceleration of the aircraft is: \[a = 1.33 \frac{\mathrm{m}}{\mathrm{s^{2}}}\]

Convert cruising velocity to meters per second

We have the cruising velocity of the aircraft as 720 km/hr. To use this value in our calculations, we need to convert it to meters per second (m/s). We can do this by first converting kilometers to meters and then hours to seconds: \[720 \frac{\mathrm{km}}{\mathrm{hr}} \times \frac{1,000 \mathrm{~m}}{1 \mathrm{~km}} \times \frac{1 \mathrm{~hr}}{3600 \mathrm{~s}}\] Simplifying the equation, we find that the cruising velocity of the aircraft in meters per second is: \[v = 200 \frac{\mathrm{m}}{\mathrm{s}}\]

Calculate the time to reach cruising velocity

Now we have velocity (v) and acceleration (a); we can use one of the equations of motion to find the time (t) it takes for the aircraft to reach its cruising velocity. Let's use the equation \(v = u + at\), where u is the initial velocity (0 m/s since the aircraft starts from rest), v is the final velocity (200 m/s), and a is the acceleration (1.33 m/s²). Rearranging the equation to find t, we have \[t = \frac{v-u}{a}\] Substituting the known values, we get: \[t = \frac{200 \mathrm{~m/s}-0 \mathrm{~m/s}}{1.33 \mathrm{~m/s^{2}}}\] Simplifying the equation, we find that the time it takes for the aircraft to reach its cruising velocity is: \[t = 150 \mathrm{~s}\] So, the correct answer is option 2: \(150 \mathrm{~s}\).

Key concepts

Physics Problem Solving

Problem solving in physics fundamentally revolves around understanding the situation, utilizing the correct physics principles, and applying mathematical techniques to solve for the unknowns. For instance, in the exercise provided, the challenge was to determine the time taken for a Boeing 737 to reach its cruising velocity. The scenario required comprehension of basic mechanics principles and the application of Newton's Second Law of Motion, which relates forces acting on an object to its acceleration.

Similarly, crucial to solving physics problems is breaking them down into manageable steps. The first step typically involves identifying the physical quantities involved and what you are solving for—in this case, time to reach a certain velocity. The second step involves translating the provided information into mathematical forms, often involving formulas. And finally, performing the necessary mathematical operations to arrive at the solution. By meticulously following these steps and cross-referencing physical concepts with mathematics, one can methodically approach and solve physics problems.

Converting Units

A common necessity in physics problem solving is converting units for consistency and correctness. Units need to be compatible with each other to accurately perform mathematical operations. In the aforementioned exercise, the aircraft’s cruising velocity was originally given in kilometers per hour (km/hr) but needed to be converted to meters per second (m/s) before we could use it with the acceleration unit (m/s²) to find the time.

The conversion process is straightforward: multiply by the ratio of the units you want to cancel to the units you need. To convert the aircraft's speed, the given velocity in km/hr was multiplied by conversion factors:

• 1000 meters in a kilometer (to convert km to m),
• 1 hour is equal to 3600 seconds (to convert hr to s).

Remember, the choice of units can vary depending on the problem's context or convention in a particular field of physics. Therefore, clarity in conversions is essential for accuracy and understanding in physics calculations.

Equations of Motion

The equations of motion are a set of four kinematic equations that describe the relationship between an object's velocity, acceleration, displacement, and time. They are central to understanding how objects move. The equation used to solve for time in reaching the aircraft's cruising velocity was one of these motion equations, specifically for a case where initial velocity ($It$ )\; is\; zero\; and\; acceleration\; is\; constant.$$

This equation, $v\; =\; u\; +\; at$ ,\; reveals\; the\; direct\; relationship\; between\; final\; velocity\; ($ v$ ),\; initial\; velocity\; ($ u$ ),\; acceleration\; ($ a$ ),\; and\; time\; ($ t$ ).\; Rearranging\; it\; allows\; us\; to\; solve\; for\; any\; of\; these\; values\; when\; the\; others\; are\; known.\; Understanding\; when\; and\; how\; to\; use\; these\; equations\; is\; pivotal\; to\; solving\; a\; wide\; range\; of\; problems\; in\; physics,\; from\; the\; simplest\; to\; the\; most\; complex\; systems,\; ensuring\; a\; strong\; foundation\; in\; describing\; motions\; of\; objects.$$$$$$$$$$