Problem 52 - College Physics

Question

At the end of most landing runways in airports, an extension of the runway is constructed using a special substance called formcrete. Formcrete can support the weight of cars, but crumbles under the weight of airplanes to slow them down if they run off the end of a runway. If a plane of mass \(2.00 \times 10^{5} \mathrm{~kg}\) is to stop from a speed of \(25.0 \mathrm{~m} / \mathrm{s}\) on a \(100-\mathrm{m}\) -long stretch of formcrete, what is the average force exerted on the plane by the formcrete?

Step-by-Step Solution

Answer

The average force exerted by the formcrete on the plane is \(-6.25 \times 10^{5} \text{ N}\).

Step 1: Understand the Problem

We need to find the average force exerted by the formcrete on the plane to bring it to a stop. This situation involves stopping the plane of mass \(2.00 \times 10^{5} \text{ kg}\) from an initial velocity \(v_i = 25.0 \text{ m/s}\) over a displacement \(d = 100 \text{ m}\).

Step 2: Apply Kinematic Equation

We use the kinematic equation to find the acceleration needed to bring the plane to rest:\[v_f^2 = v_i^2 + 2a d\] where \(v_f = 0 \text{ m/s}\) (final velocity), \(v_i = 25.0 \text{ m/s}\), \(d = 100 \text{ m}\), and solve for \(a\).

Step 3: Solve for Acceleration

Substitute the known values into the equation: \(0 = (25.0)^2 + 2a (100)\). Simplify and solve for \(a\):\[0 = 625 + 200a\]\[-625 = 200a\]\[a = -625/200 = -3.125 \text{ m/s}^2\].

Step 4: Use Newton's Second Law to Find Force

Using the calculated acceleration, we apply Newton's Second Law: \(F = ma\), where \(m = 2.00 \times 10^{5} \text{ kg}\) and \(a = -3.125 \text{ m/s}^2\). Plug these values in to calculate the force.

Step 5: Calculate the Average Force

Calculate: \[F = (2.00 \times 10^{5}) \times (-3.125) = -6.25 \times 10^{5} \text{ N}\]. The negative sign indicates the force direction is opposite to the plane's motion, stopping it.

Key Concepts

Kinematic Equations

Kinematic equations are essential in describing motion. They relate the different aspects of motion, such as displacement, initial velocity, final velocity, time, and acceleration. In this exercise, we use one such kinematic equation:

• \(v_f^2 = v_i^2 + 2a d\)

This equation allows us to calculate the acceleration of the plane when it is stopping. By substituting the values given, such as the initial speed, final speed (which is zero in this case), and the distance covered, we find the acceleration needed to bring the plane to a stop.Understanding these equations helps solve many physics problems involving motion without the need to know the time taken for the motion.

Newton's Second Law

Newton's Second Law of Motion is a critical principle that explains how the velocity of an object changes when it is subjected to an external force. The law is summarized by the equation:

• \(F = ma\)

Where:

• \(F\) is the force applied,
• \(m\) is the mass of the object,
• \(a\) is the acceleration.

In this scenario, after determining the acceleration using the kinematic equations, we apply Newton's Second Law to calculate the force. This force is how much the formcrete must exert to stop the plane. This understanding is vital in many real-life situations where stopping a moving object safely and effectively is necessary.

Mass and Acceleration

Mass and acceleration are key elements in determining the force needed to change the motion of an object. Mass refers to the quantity of matter in an object, and it plays a crucial role in how much force is needed to change the object's speed or direction.Acceleration, on the other hand, is the rate at which the velocity of an object changes. It's measured in meters per second squared (\(m/s^2\)).When calculating force, understanding the difference between these two aspects is critical:

• Heavier objects (more mass) require more force to accelerate.
• The more acceleration you want, the more force you need.

This relationship shows why airplanes, which are massive, require substantial forces to cause any quick changes in motion, such as stopping or taking off.

Stopping Distance

Stopping distance is the total distance required for a moving object to come to a stop. It is influenced by the initial speed of the object, the mass, and the forces acting against its motion, such as friction or, in this case, the force applied by the formcrete. For the plane landing on formcrete:

• The initial velocity is given.
• We calculate the stopping distance based on the force applied through kinematic equations and Newton's laws.

Understanding stopping distance is crucial in designing systems or materials, like formcrete, to safely bring moving objects to rest. It's a vital component in ensuring transportation safety, particularly in high-speed environments like airports.

Runway Safety

Runway safety is a significant concern in aviation, ensuring that aircraft can land and take off safely under all conditions. Various features, such as formcrete, are employed at the runway's end to enhance safety. These features are designed to stop planes quickly without causing damage or risks to passengers and crew. Key considerations in runway safety include:

• Creating adequate stopping distance for aborted takeoffs or landings.
• Using materials that safely absorb and dissipate energy, like formcrete.
• Ensuring that the runway surface can support occasional vehicle movement when necessary.

Safety measures like these are essential in maintaining air travel as one of the safest modes of transportation available. Understanding how these systems work can deepen appreciation for the complex measures taken to secure passengers aboard airplanes.