In contrast to Problem 16 , consider the perfectly elastic bouncing of a ball vertically under gravity above the plane \(x=0\). We have \(\dot{x}=y, \dot{y}=\) \(-g\) and we can solve for \(x\left(x_{0}, y_{0}, t\right), y\left(x_{0}, y_{0}, t\right)\) in terms of the initial data \(\left(x_{0}, y_{0}\right)\) at \(t=0\). Show that in this case the resulting perturbations \(\Delta x, \Delta y\) essentially grow linearly with time \(t\) along the trajectory when we make perturbations \(\Delta x_{0}, \Delta y_{0}\) in the initial data. That is to say the distance along the trajectory \(d \equiv \sqrt{(\Delta x)^{2}+(\Delta y)^{2}} \sim \kappa t\) when \(t\) is large and \(\kappa\) is a suitable constant.

When studying the movement of objects, especially under the influence of gravity, it's crucial to understand the equations of motion. These equations allow us to predict the future position and velocity of an object from its current state. For an object moving under gravity, like a bouncing ball, the vertical position and the vertical velocity are usually represented by the variables x and y, respectively.

The change of position with respect to time, also known as velocity, is denoted by \(\dot{x}\). Similarly, the change of velocity with respect to time, or acceleration, is denoted by \(\dot{y}\). For an object in free-fall under gravity without air resistance, the only acceleration acting on it is due to gravity (\(g\)), which is a constant value directed downwards. Thus, we obtain the equations \(\dot{x} = y\) and \(\dot{y} = -g\), where the negative sign represents the direction opposite to the chosen positive coordinate direction.

The concept of initial conditions is vital for solving equations of motion. Initial conditions specify the state of an object at the starting point of observation, typically at time \(t = 0\). In this exercise, the initial conditions are the initial position \(x_0\) and the initial velocity \(y_0\) of the bouncing ball.

These conditions serve as the necessary input to solve the equations of motion and determine the object's trajectory. Without initial conditions, the general solution to the equations of motion would encompass an infinite number of possible trajectories. By setting \(x_0\) and \(y_0\), we are able to find the unique path that the object will follow.

The idea of trajectory perturbation involves slightly altering the initial conditions of an object's movement and observing how these alterations change the path it takes. This concept is not only fascinating for theoretical physics but also for understanding real-world systems that are sensitive to initial conditions.

In our scenario, perturbations \(\Delta x_0\) and \(\Delta y_0\) are introduced to the initial position and velocity. This change leads to a deviation in the trajectory of the ball from its original path. By calculating the differences in the \(x\) and \(y\) values due to these perturbations, one can analyze the stability of the system and predict how errors or uncertainties in measurements could affect the outcome.

When we delve into linear growth perturbations, we're examining how small changes in the initial conditions can lead to variations that increase or decrease at a rate proportional to time. In our ball bouncing problem, the perturbations in position and velocity (\(\Delta x\) and \(\Delta y\)) scale with time \(t\), indicative of a linear relationship.

For large values of time, we find that the deviations from the original path, or the distance \(d\), can be approximated by a linear relationship \(d(t) \sim \kappa t\), where \(\kappa\) is a constant dependent on the perturbation in the velocity \(\Delta y_0\). This linear growth rate implies a predictable and proportionate divergence from the expected trajectory over time, which is a fundamental aspect of understanding motion and stability in physical systems.