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In the realm of physics, vertical motion refers to the movement of objects in a straight line along the vertical axis. It's a crucial concept for understanding projectile motion, particularly when objects are moving under the influence of gravity alone. In this context, arrows shot vertically are impacted by gravity, pulling them downward at a constant rate of approximately 9.8 meters per second squared. This force is denoted as 'g' in calculations.
When an object is launched upward, it battles the force of gravity, causing its speed to decrease as it ascends. Eventually, it will reach its peak height – the point where its velocity is momentarily zero before it begins its descent back to the ground. With this understanding, one can see how the initial speed and time of launch significantly influence vertical motion. Analyzing such motion requires a solid grasp of kinematics equations.

Kinematics equations are mathematical formulas that describe the motion of objects without considering the forces causing them. These equations are fundamental in solving problems related to vertical motion in physics. There are three key kinematic equations, each assisting in different scenarios involving velocity, time, and displacement. The primary equation used here tackles calculating the time it takes for an object to reach its maximum height.
When dealing with vertical motion, particularly with objects thrown upwards, the time to reach the maximum height can be determined using the formula:

• \[ t_{max} = \frac{v_0}{g} \]

Here, \(v_0\) is the initial velocity, and \(g\) is the acceleration due to gravity. This equation represents how the starting speed impacts the time required to reach the peak height when an object moves upwards. Understanding these equations enables us to solve complex problems, like the one tackled in the original exercise.

Calculating the initial speed of an object in motion is essential for predicting its future position or state. In the context of our problem, identifying the second arrow's initial speed ensures both arrows reach their peak heights simultaneously, despite their different launch times. This calculation hinges upon the concept that the remaining time for the second arrow must be calculated by considering the delay in launch.
The formula connecting initial speed to time elapsed and gravitational force is:

• \[ v_0 = g \cdot t_{max} \]

Using this formula enables us to back-calculate the necessary initial speed when the time to reach the maximum height is known. As shown in the solution, substituting the known gravitational constant \(9.8 \, \text{m/s}^2\) and the calculated time \(1.35 \, \text{s}\) provides the initial speed \(13.23 \, \text{m/s}\) for the second arrow. Understanding how to manipulate these equations is crucial for finding unknown variables in vertical motion scenarios.