Question

A jet touches down on a runway with a speed of \(142.4 \mathrm{mph}\). After \(12.4 \mathrm{~s},\) the jet comes to a complete stop. Assuming constant acceleration of the jet, how far down the runway from where it touched down does the jet stand?

Short answer

Answer: The jet traveled approximately 393.55 meters down the runway before coming to a complete stop.

Step-by-step solution

Convert speed and time into SI units

First, convert the initial speed from mph to m/s and the time from seconds to hours. 1 mph = 0.44704 m/s and Since the time is given in seconds, we don't need to convert it. Initial speed, \(v_0 = 142.4 \times 0.44704 = 63.669696 \mathrm{~m/s}\) Time, \(t = 12.4 \mathrm{~s}\)

Calculate the acceleration

To calculate the acceleration, we can use the kinematic equation: $$v = v_0 + at$$ where, \(v\) is the final velocity \(v_0\) is the initial velocity \(a\) is the acceleration \(t\) is the time Since the jet comes to a complete stop, its final velocity is 0. Let's solve for acceleration, \(a\): $$a = \frac{v - v_0}{t}$$ \(a = \frac{0 - 63.669696}{12.4} = -5.140{}467 \mathrm{~m/s^2}\)

Calculate the distance

Now, we will use another kinematic equation to find the distance the jet traveled during this process: $$s = v_0t + \frac{1}{2}at^2$$ where, \(s\) is the distance Substitute the given values and solve for \(s\): \(s = 63.669696 \times 12.4 + \frac{1}{2}(-5.140467) \times 12.4^2\) \(s = 790.2307392 - 396.6772932 = 393.553446 \mathrm{~m}\) The jet traveled approximately 393.55 meters down the runway before coming to a complete stop.

Key concepts

Physics Problem Solving

Physics problems often seem daunting, but breaking them down into manageable steps is key to finding a solution. In the case of our jet on the runway, the problem first asks us to find the distance the jet travels before it comes to a stop. The process begins with understanding the problem statement and identifying the knowns and unknowns. Here, we know the jet's initial speed, the time it takes to stop, and that the acceleration is constant. The unknown is the distance the jet travels, denoted as s.

Effective problem solving in physics also involves choosing the right equations. We use the kinematic equations because they relate the variables of motion (distance, speed, acceleration, and time) under the assumption of constant acceleration. Always strive to simplify the problem, such as by considering one dimension of motion, and to keep an eye out for any conversions needed, as units must be consistent throughout the problem. Furthermore, making a sketch can often help visualize the situation, which is a powerful tool in physics problem solving.

Constant Acceleration

The concept of constant acceleration is central to kinematic problems in physics. It refers to a consistent change in velocity over time. In our jet example, assuming constant deceleration means that the jet slows down at a regular rate as it travels down the runway. This assumption simplifies calculations and allows us to use certain kinematic equations safeguardedly.

Remember, acceleration can be positive or negative. Positive acceleration indicates an increase in velocity, while negative acceleration, or deceleration, indicates a decrease. In our jet problem, the acceleration is negative because the jet is coming to a stop. The value of acceleration, a, is found by rearranging the first kinematic equation, v = v_0 + at, which tells us how velocity changes over time when acceleration is constant. Constant acceleration allows us to predict the motion of objects with great accuracy, which is why it's a staple in motion analysis.

Unit Conversion

Working with units correctly is an essential skill in physics and many other sciences. Incorrect unit conversion can lead to entirely wrong answers. As in our jet problem, we started with the speed in miles per hour (mph) but needed to convert to meters per second (m/s) because SI units are the standard in physics equations. The conversion factor 1 mph = 0.44704 m/s is crucial.

Unit conversion involves multiplying the original measurement by a conversion factor that represents a ratio equal to one. This does not change the value, just the units. Often, remembering key conversion factors or knowing where to find them is part of the problem-solving process. Also, keeping track of your units throughout calculations serves as a valuable check that your process is on the right track. Once all units in your problem are consistent, you can proceed with solving for the unknown using the appropriate formulas.