Question

An airplane starts from rest and accelerates at \(12.1 \mathrm{~m} / \mathrm{s}^{2}\). What is its speed at the end of a \(500 .-\mathrm{m}\) runway?

Short answer

Answer: The final speed of the airplane is approximately 110 m/s.

Step-by-step solution

Write the given information

We are given the following information: - Initial velocity, \(u = 0 \mathrm{~m/s}\) (since the airplane starts from rest) - Acceleration, \(a = 12.1 \mathrm{~m/s^2}\) - Distance along the runway, \(s = 500 \mathrm{~m}\)

Use the kinematics equation for acceleration and distance

Use the kinematics equation, which is \(v^2 = u^2 + 2as\). Substitute the given information: \(v^2 = 0 + 2(12.1)(500)\)

Calculate the final speed squared

Calculate the final speed squared: \(v^2 = 2(12.1)(500) = 12100\)

Find the final speed

Since we have found \(v^2\), we can now find the final velocity by taking the square root of both sides: \(v = \sqrt{12100} \approx 110 \mathrm{~m/s}\) Therefore, the speed of the airplane at the end of the \(500\mathrm{-m}\) runway is approximately \(110 \mathrm{~m/s}\).

Key concepts

Uniformly Accelerated Motion

Uniformly accelerated motion refers to an object that is increasing its velocity by a consistent amount each second, effectively having a constant acceleration. It's crucial to understand this concept as it applies to bodies moving in a straight line when the acceleration does not change over time. When an airplane accelerates along a runway, like in our exercise, it is generally assumed to be experiencing uniformly accelerated motion, barring factors like air resistance.

Understanding this uniform acceleration is foundational, as it simplifies the calculations involved in motion analysis. With a consistent acceleration value, such as the given \(12.1 \mathrm{~m/s^2}\), we can predict future velocity and position using kinematic equations. When we say an airplane starts from rest, it simply means that its initial velocity was zero, making calculations even simpler.

Final Velocity Calculation

Calculating the final velocity of an object is essential to understanding its motion at the end of its acceleration period. The final velocity is what the object reaches after a certain distance or time at a given acceleration, starting from an initial velocity. For instance, in the given exercise, we wanted to find the velocity of an airplane after it has traveled the length of a 500-meter runway.

To calculate this velocity, we use the fact that the initial velocity ( u) was zero because the airplane started from rest. By applying the kinematic equation \(v^2 = u^2 + 2as\), where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(s\) is the distance, we could easily solve for \(v\) the final speed of the airplane on the runway.

Kinematic Equations for Acceleration

Kinematic equations are the mathematical formulas used to describe the motion of objects in terms of their velocity, acceleration, and position over time. These equations become immensely useful when dealing with uniformly accelerated motion, as they allow you to relate these varying quantities to each other.

For acceleration calculations, one of the most commonly used kinematic equations is \(v^2 = u^2 + 2as\), which relates final velocity to initial velocity, acceleration, and displacement. This particular equation is independent of time, which means it's helpful when time is not known or is not a required variable. In our exercise example, the kinematic equation allowed us to bypass the need to consider time and directly calculate the final velocity based on the known acceleration and distance.