Question

A jumbo jet needs to reach a speed of $360\mathrm{km}/\mathrm{h}(=224\mathrm{mi}/\mathrm{h})$ on the runway for takeoff. Assuming a constant acceleration and a runway $1.8\mathrm{km}(=1.1\mathrm{mi})$ long, what minimum acceleration from rest is required?

Short answer

The minimum acceleration required for the jumbo jet to take off can be calculated from the motion equation. After converting to suitable units and substituting the given values, it can be calculated by dividing the difference of squares of final and initial speeds by twice the distance.

Step-by-step solution

Convert units to SI

Before substituting values into the equation, make sure all are in appropriate units. Convert speed from kilometers per hour to meters per second and distance from kilometers to meters. Use the conversions 1 km = 1000 m and 1 hour = 3600 seconds. Thus, final speed v = $\frac{360\times 1000}{3600}$ m/s and distance s = $1.8\times 1000$ m.

Substitute into equation and solve

Substitute the values into the equation $v^2=u^2+2as$. Here, final speed v is the converted value, initial speed u = 0 (as the jet starts from rest), and distance s is the converted value. Rearrange the equation to solve for acceleration a = $\frac{v^2-u^2}{2s}$.

Compute acceleration

After substituting values into the equation, perform the computation and round the result to a suitable number of significant figures. This will give the minimum required acceleration for the jumbo jet to take off.

Key concepts

Constant Acceleration

Constant acceleration is a fundamental concept in kinematics, which deals with the motion of objects. When an object accelerates at a constant rate, its speed increases or decreases steadily over time, which means its acceleration stays the same throughout the motion period.
This is common in scenarios like when a jumbo jet takes off since the engines provide a steady increase in speed until the plane reaches the necessary velocity to lift off the ground.
The simplicity of constant acceleration allows us to predict and calculate various aspects of motion using specific formulas.

Conversion of Units

Before solving physics problems, it's important to have parameters in consistent units, often necessitating a conversion of units. For instance, in the context of airplane takeoff, speed might be given in kilometers per hour, while the runway length is provided in kilometers.
To apply kinematic equations effectively, we convert these into meters per second and meters respectively. The conversion involves multiplying kilometers by 1000 to get meters and dividing kilometers per hour by 3.6 to convert it to meters per second.

• 1 km = 1000 meters
• 1 km/h = 1000 meters / 3600 seconds = 1/3.6 m/s

This ensures all values align with the standard metric units used in kinematic equations.

Equations of Motion

Equations of motion are mathematical formulas used to predict the future position and velocity of objects moving with constant acceleration. These are particularly useful in analyzing the takeoff speed of airplanes.
The primary equation referenced in our example is given by: $$v^2=u^2+2as$$where:

• $v$ is the final velocity,
• $u$ is the initial velocity,
• $a$ is the acceleration,
• and $s$ is the displacement.

This formula helps determine one variable if the others are known, such as finding acceleration when the final velocity and displacement are known. Rearranging terms allows for solving the equation specific to the missing variable you desire.

Airplane Takeoff

In aviation, understanding kinematics is essential for calculating the performance requirements of takeoff. Airplanes, including jumbo jets, need to reach a specific speed before they can take off. This speed depends on factors such as aircraft weight, air pressure, and wind conditions.
On the runway, the plane accelerates from rest, requiring a certain distance to reach the necessary takeoff velocity. By applying constant acceleration, pilots and engineers use kinematics to ensure the runway is long enough for the aircraft to gain sufficient speed.
This ensures a safe and effective takeoff, optimizing performance and maintaining safety standards.