Question

The Great Zacchini, daredevil extraordinaire, is a circus performer whose act consists of being "shot from a cannon" to a safety net some distance away. The "cannon" is a frictionless tube containing a large spring, as shown in the figure. The spring constant is \(k=150 \mathrm{lb} / \mathrm{ft}\), and the spring is precompressed \(10 \mathrm{ft}\) prior to launching the acrobat. Assume that the spring obeys Hooke's law and that Zacchini weighs \(150 \mathrm{lb}\). Neglect the weight of the spring. (a) Let \(x(t)\) represent spring displacement along the tube axis, measured positive in the upward direction. Show that Newton's second law of motion leads to the differential equation \(m x^{\prime \prime}=-k x-m g \cos (\pi / 4), x<0\), where \(m\) is the mass of the daredevil. Specify appropriate initial conditions. (b) With what speed does he emerge from the tube when the spring is released? (c) If the safety net is to be placed at the same height as the mouth of the "cannon," how far downrange from the cannon's mouth should the center of the net be placed?

Short answer

Answer: The final step is to find the horizontal distance traveled by The Great Zacchini when he reaches the same height as the mouth of the cannon by using the horizontal component of the velocity and the time it takes to reach that height. This horizontal distance will determine the position of the safety net.

Step-by-step solution

Determine the forces acting on Zacchini and derive the differential equation from Newton's Second Law

Newton's second law states that \(F = ma\), where \(F\) is the force acting upon an object, \(m\) is its mass, and \(a\) is its acceleration. In this problem, two forces are acting on Zacchini: the force from the spring (\(F_s\)) and gravity (\(F_g\)). The spring force obeys Hooke's law, which states \(F_s = -kx\), where \(k\) is the spring constant and \(x(t)\) is the spring displacement along the tube axis. The gravitational force is given by \(F_g = -mg \cos(\pi / 4)\) because of the inclined angle, where \(m\) is the mass of Zacchini and \(g\) is the acceleration due to gravity. We know that \(F = ma = F_s + F_g\). Rewriting this equation, we get: $$m x^{\prime \prime} = -kx - mg \cos(\pi / 4), \quad x < 0$$ where \(x^{\prime \prime}\) is the acceleration. We are also given the mass of the daredevil, \(m = 150 \text{lb} / 32.174 \text{ft/s}^2 =\) 4.667 slugs, since 1 slug is equal to 32.174 \(\text{lb}/\text{ft/s}^2\). Now, we have the differential equation: $$4.667 x^{\prime \prime} = -150 x - 4.667g \cos(\pi / 4), \quad x < 0$$

Specify the initial conditions

We need to specify the initial conditions in order to solve the differential equation. When the spring is initially compressed 10 ft (\(x = -10\)), Zacchini is at rest (\(x^{\prime}(0) = 0\)). The initial conditions are, therefore: $$x(0) = -10 \text{ft} \quad \text{and} \quad x^{\prime}(0) = 0 \text{ft/s}$$

Solve the differential equation to find the speed when the spring is released

To solve this problem, let's convert the differential equation into a second order homogeneous linear equation by dividing by the mass (\(m\)) and adding the constant term: $$x^{\prime \prime} + \frac{k}{m}x = -g \cos(\pi / 4)$$ This equation can be solved using standard techniques for second order linear homogeneous differential equations as follows: Let a new variable \(y\) be equal to the velocity of Zacchini, i.e., \(y = x^{\prime}\). Then, we have: $$y^{\prime} + \frac{k}{m}y = -g \cos(\pi / 4)$$ Now we want to solve this first order linear inhomogeneous differential equation for \(y(t)\): $$y^{\prime} + \frac{k}{m}y = -g \cos(\pi / 4)$$ and apply the initial condition \(y(0) = x^{\prime}(0) = 0\). After solving this equation, we can find \(y(t)\), and the speed at which Zacchini emerges from the tube is given by \(y(t)\) when \(x(t) = 0\).

Find the position of the safety net

Using the speed found in Step 3, we want to calculate the trajectory of Zacchini. Since the cannon is inclined at 45 degrees (\(\pi / 4\) radians), we can break down the velocity into its horizontal and vertical components. The horizontal component of the velocity will remain constant throughout the entire trajectory, while the vertical component will change due to the acceleration due to gravity. Using these velocity components, we can find the time when Zacchini again reaches the same height as the mouth of the cannon (i.e., when the net should be placed). Using this time, we can then find the horizontal distance traveled and determine the position of the safety net.

Key concepts

Newton's second law of motion

Newton's second law of motion is a fundamental principle that underpins much of classical mechanics. It tells us how the motion of an object changes when it is subjected to external forces. The law is usually expressed with the equation
\[ F = ma \]
where \(F\) denotes the net force applied to the object, \(m\) is the object's mass, and \(a\) is its acceleration. This relationship is crucial when analyzing problems in physics, as it links the dynamics of an object to the forces acting upon it.
In the context of our daredevil Zacchini being shot from a spring-powered cannon, this law helps us derive a differential equation that models his motion. As the spring releases its stored energy, it applies a force on Zacchini that propels him along the tube's axis. By incorporating both the spring force and gravity's component along the inclined path, Newton's second law enables us to predict his acceleration at any point during the launch.

Hooke's law

Hooke's law is a principle of physics that relates the force exerted by a spring to the distance it is stretched or compressed from its resting position. It can be represented as:
\[ F_s = -kx \]
This equation tells us that the spring force (\(F_s\)) is directly proportional to the displacement (\(x\)) of the spring from its equilibrium position, with \(k\) being the spring constant, a measure of the spring's stiffness. The negative sign indicates that the force exerted by the spring is always in the opposite direction of the displacement.
We apply Hooke's law to our example of Zacchini, where the 'cannon' spring is compressed, storing potential energy. Upon release, this energy converts into kinetic energy, propelling Zacchini through the air. Understanding this law is essential for determining the initial force acting on Zacchini and the subsequent motion he experiences.

Initial conditions in differential equations

In the world of differential equations, initial conditions play a pivotal role in defining a unique solution to a given problem. These conditions are the values of the unknown function and its derivatives at a specific point, usually when time \(t=0\).
Using initial conditions, we can pin down the particular solution to a differential equation that satisfies our problem's specific physical scenario. For the acrobat Zacchini, the initial conditions represent the starting point of his journey when the spring is fully compressed, and he has not yet begun moving. This helps us determine his position and velocity at the moment the spring is released, setting the stage for the rest of his flight to the safety net.

Homogeneous linear differential equations

Homogeneous linear differential equations are a class of differential equations that can be characterized by the absence of a term devoid of the function (or its derivatives) we're solving for. When faced with such an equation, the solution process usually involves finding the characteristic equation, solving for the roots, and then using them to construct the general solution.
In our textbook problem, we're not dealing with a purely homogeneous equation, since there's a non-zero term representing gravity's effect on Zacchini. However, by incorporating this gravitational term properly, we can still use techniques related to homogeneous linear differential equations to find a particular solution to the problem that satisfies our initial conditions and thus model the trajectory of Zacchini accurately.

Projectile motion

Projectile motion is a form of motion experienced by an object that is launched into the air and is subject to the force of gravity. It's an important topic in physics since it models the paths of everything from basketballs to missiles. In the absence of air resistance, the only force acting on a projectile is gravity, which affects the vertical component of the motion.
In Zacchini's case, we split the velocity with which he exits the tube into horizontal and vertical components since he is projected at an angle. The horizontal component of velocity remains constant, while the vertical component is influenced by gravity, causing Zacchini to follow a parabolic trajectory. Understanding projectile motion allows us to calculate where the safety net should be placed to ensure Zacchini lands safely after being launched from the cannon-like spring device.