An airplane's takeoff speed is \(320 \mathrm{km} / \mathrm{h}\). If its average acceleration is \(2.9 \mathrm{m} / \mathrm{s}^{2},\) how much time is it accelerating down the runway before it lifts off?

Understanding uniform acceleration is crucial for analyzing the motion of objects moving at a constant rate of acceleration, such as an airplane during takeoff. In physics, uniform acceleration refers to the consistent increase or decrease in velocity of an object over time. This means the object's speed changes by the same amount every second. For example, if an airplane accelerates down a runway at a uniform acceleration of \(2.9 \text{ m/s}^2\), it's speed increases by \(2.9 \text{ meters per second}\) every second until it reaches takeoff speed.

An important characteristic of uniform acceleration is that the motion can be accurately described and predicted using kinematic equations. These equations allow us to connect displacement, initial velocity, final velocity, acceleration, and time. When the acceleration is uniform, calculations simplify because we don't have to account for changes in acceleration over time, making problems more straightforward to solve.

Kinematic equations are fundamental tools in physics for understanding the motion of objects under uniform acceleration. There are four kinematic equations that describe different aspects of this motion, but the one most relevant to our airplane example is \(v = u + at\). This equation relates the final velocity \(v\) of the object, the initial velocity \(u\), the acceleration \(a\), and the time \(t\) it took to reach that final velocity.

For the airplane example, the initial velocity \(u\) is zero, since it starts from rest. The final velocity \(v\) needs to be determined from the takeoff speed, and the acceleration \(a\) is provided. By isolating time \(t\), we can find out how long it takes for the airplane to reach the final velocity necessary for takeoff. Through these equations, we understand not only how long the runway needs to be but also how various parameters like acceleration affect takeoff time.

In physics problems, unit conversion is a fundamental skill because equations require consistent units to work correctly. In our problem, the airplane's takeoff speed is given in \(\mathrm{km/h}\), while acceleration is in \(\mathrm{m/s}^2\), so to solve the problem, we need these units to match. More generally, converting units ensures meaningful comparisons and accurate results when working with physical quantities.

To convert speed from \(\mathrm{km/h}\) to \(\mathrm{m/s}\), we multiply by the fraction \(\frac{1000 \text{ m}}{3600 \text{ sec}}\), reflecting the number of meters in a kilometer and the number of seconds in an hour. This step is essential because if we did not convert the units, using the kinematic equations would produce incorrect answers for the time of acceleration. Remembering to perform unit conversions before calculating can prevent many common mistakes in physics problems.