Problem 14 - College Physics

Question

A loaded Boeing 747 jumbo jet has a mass of \(2.0 \times 10^{5} \mathrm{~kg} .\) What net force is required to give the plane an acceleration of \(3.5 \mathrm{~m} / \mathrm{s}^{2}\) down the runway for takeoffs?

Step-by-Step Solution

Answer

The net force required is \( 7.0 \times 10^5 \, \mathrm{N} \).

Step 1: Understand the Formula

The net force required to accelerate an object can be determined using Newton's Second Law of Motion, which is expressed by the formula \( F = ma \), where \( F \) is the net force, \( m \) is the mass, and \( a \) is the acceleration.

Step 2: Identify the Given Values

From the problem, we know the mass \( m \) of the Boeing 747 is \( 2.0 \times 10^5 \) kg and the acceleration \( a \) is \( 3.5 \, \mathrm{m/s^2} \).

Step 3: Substitute the Values into the Formula

Substitute the given values into the equation \( F = ma \):\[F = (2.0 \times 10^5 \, \mathrm{kg}) \times (3.5 \, \mathrm{m/s^2})\]

Step 4: Calculate the Net Force

Calculate the product:\[F = 7.0 \times 10^5 \, \mathrm{N}\]Thus, the net force required to give the plane the specified acceleration is \( 7.0 \times 10^5 \, \mathrm{newtons} \).

Key Concepts

Net Force Calculation

When it comes to calculating net force, Newton's Second Law of Motion acts as our guiding principle. This law shows us how to find out the net force needed to move an object with a certain amount of acceleration. Imagine you have a shopping cart. If you push it gently, it doesn't move much, but if you push it hard, it zooms away. This change in speed is what we call acceleration. The formula to calculate net force is simple:

• \( F = ma \)

Here, \( F \) represents the net force in newtons (N), \( m \) is the mass in kilograms (kg), and \( a \) stands for acceleration in meters per second squared (m/s²).
For example, in the situation with the Boeing 747, understanding this law means knowing that each component term in \( F = ma \) has a rightful place and purpose.
All we need to do is plug in the mass of the plane, which is given as \( 2.0 \times 10^5 \) kg, and its acceleration \( 3.5 \, \mathrm{m/s^2} \), into this formula to get the net force.
This calculation provides clarity on how much force is needed to make massive objects, like a plane, take off smoothly.

Mass and Acceleration Relation

The relationship between mass and acceleration is fundamental to understanding motion dynamics in physics. Mass is essentially how much matter there is in an object. It doesn't change whether you're on Earth, the Moon, or Mars. In contrast, acceleration is how fast an object speeds up or slows down.
Newton's Second Law beautifully ties these two together. It tells us that the force needed to accelerate an object grows if either the mass or the acceleration increases. Let's break it down:

• If the mass stays constant and you want the object to accelerate faster, you'll need to apply a greater force.
• Alternatively, with the same force, a lighter object will accelerate more than a heavier one.

In our Boeing 747 problem, the mass is quite large— \( 2.0 \times 10^5 \) kg—showing you why such substantial force is required for a noticeable acceleration of \( 3.5 \mathrm{~m/s^2} \).
This interdependence of mass and acceleration through force is why heavier vehicles, like airplanes or trucks, need much more powerful engines compared to small cars.

Physics Problem Solving

Solving physics problems, especially those involving motion, might feel overwhelming at first, but breaking them into steps makes the process manageable. Just like assembling a puzzle, each step aligns pieces of the larger picture.
Let's explore some easy-to-remember steps:

• First, identify what you are asked to find. In our case, it is the net force.
• Next, list out the known values. This includes both mass and acceleration here.
• Apply the relevant formula—in this situation, Newton’s Second Law, \( F = ma \).
• Input the values into the formula and compute.
• Review the solution to ensure that the units and calculations make sense.

Methodically working through each step gives clarity and builds confidence.
The Boeing 747 example illustrates this problem-solving approach: calculating \( 7.0 \times 10^5 \, \mathrm{N} \) as the force needed demonstrates how following these steps leaves little room for error, providing a structured path to the solution.