Question

A 26-ton Navy jet (Fig. 3-27) requires an air speed of \(280 \mathrm{ft} / \mathrm{s}\) for lift-off. Its own engine develops a thrust of \(24,000 \mathrm{lb}\). The jet is to take off from an aircraft carrier with a \(300-\mathrm{ft}\) flight deck. What force must be exerted by the catapult of the carrier? Assume that the catapult and the jet's engine each exert a constant force over the 300 -ft takeoff distance.

Short answer

The catapult must exert a force of approximately \(186927.57 lb\).

Step-by-step solution

Determine the Acceleration Required

We can use the kinematic equation \(v^2 = u^2 + 2as\) where \(v\) is the final speed (280 ft/s), \(u\) is the initial speed (0 ft/s), \(a\) is the acceleration, and \(s\) is the distance (300 ft). Solving for \(a\) gives \(a = (v^2 - u^2) / (2s)\). Substituting the known values, we get \(a = (280^2 - 0) / (2 * 300) = 130.67 ft/s^2.

Calculate the Total Force Needed

We use the formula \(F = ma\), where \(m\) is the mass of the jet and \(a\) is the acceleration determined in the previous step. The mass needs to be converted to slug (mass unit in the Imperial system) from tons for unit consistency. 1 ton = 2000 lb, and 1 slug = 32.2 lb. So, \(m = 26 * 2000 / 32.2 = 1615.53 slug\). Substituting the values in the equation gives \(F = 1615.53 * 130.67 = 210927.57 lb\). This is the total force required for the jet to lift off.

Find the Force Exerted by the Catapult

The total force required for lift-off is the sum of the forces exerted by the jet engine and the catapult. To find the force exerted by the catapult, we subtract the force from the jet engine from the total force required. So, the force exerted by the catapult is \(210927.57 lb - 24000 lb = 186927.57 lb\).

Key concepts

Kinematics

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause that motion. It involves several key concepts and equations which allow us to determine quantities such as velocity, acceleration, displacement, and time.

• Velocity is the rate of change of an object’s position. It tells us how fast an object is moving and in what direction.
• Acceleration is the rate of change of velocity. When an object speeds up, slows down, or changes direction, it is accelerating.
• Displacement is the change in position of an object. Unlike distance, it is a vector quantity, which means it has both magnitude and direction.

In the context of the exercise, the kinematic equation \(v^2 = u^2 + 2as\) is used to find the required acceleration for the jet to reach take-off speed. Here, \(v\) is the final velocity (280 ft/s), \(u\) is the initial velocity (0 ft/s), \(a\) is acceleration, and \(s\) is the takeoff distance (300 ft). This equation is handy when you need to connect these variables without direct consideration of time.

Force and Motion

In physics, understanding force and motion is crucial, as they are directly related. Forces are pushes or pulls that can cause an object to move, stop, or change direction. According to Newton's Second Law, the force acting on an object is equal to the mass of that object times its acceleration (\(F = ma\)).

To solve the given exercise, it’s important to consider:

• The total force needed to accelerate the jet to the required speed for take-off.
• The thrust provided by the jet's own engine.
• The additional force required from the aircraft carrier’s catapult.

Here, the total force is calculated using the jet's mass in slugs and the required acceleration over a specific distance.
The equation \(F = ma\) helps us to quantify these forces to ensure that the jet reaches its take-off speed.

Newton's Laws

Understanding Newton's Laws is fundamental in solving physics problems related to force and motion. Newton's laws consist of three main principles that describe how objects move.

For this specific problem:

• Newton’s First Law (Law of Inertia) states that an object will remain at rest or continue at a constant velocity unless acted upon by a net external force.
• Newton’s Second Law formulates the relation as \(F = ma\), indicating that force is the product of mass and acceleration. This is the key law used in our exercise to determine the force required for the jet's takeoff.
• Newton’s Third Law states that for every action, there is an equal and opposite reaction. Although not directly used in calculation, it explains the interaction between the jet and the catapult.

The solution to the exercise relies heavily on Newton's Second Law, as it provides a method to calculate the force that the jet engine and the catapult must supply to achieve the desired take-off speed.