91Ó°ÊÓ

In physics, kinematic equations are essential for predicting projectile motion. These equations help to describe the motion of objects under the influence of constant acceleration. For projectile motion, this constant acceleration is usually gravity, acting downwards. Here are some of the key kinematic equations used in such problems:

• Velocity-Position Relation: \(v^2 = v_0^2 + 2a (x - x_0)\) - This is used to find the velocity at any point given the initial velocity and acceleration.
• Position-Time Relation: \(x = x_0 + v_0 t + \frac{1}{2} a t^2\) - Useful for finding an object's position at any given time.
• Velocity-Time Relation: \(v = v_0 + at\) - Used to find an object's velocity at any given time.

In the context of our exercise, we specifically used the velocity-position relation to find the maximum height of the baseball and its velocity at a given height during its descent.

Initial velocity is the speed of an object at the start of its motion. In many projectile motion problems, this is a critical value because it determines how far and how high the object will go. For the given problem, the baseball is thrown with an initial velocity (\(v_0\)) of 100 ft/s straight up.
Understanding initial velocity helps establish the base conditions for solving further kinematic equations. Knowing this value allows us to calculate aspects such as how long the object will be in the air or, as in our calculation, how high the ball will ascend against gravity.

Acceleration due to gravity is a constant force affecting all objects in free fall near the Earth's surface. It pulls objects toward the Earth at approximately \(32 \text{ ft/s}^2\) (in the downward direction in customary units).
This negative acceleration is crucial in determining how quickly objects slow down when moving upwards and speed up when coming down. In our baseball example, gravity was considered to be a force acting against the initial velocity as the ball rises, eventually bringing its speed to zero at its peak height before pulling it back down again.

The maximum height of a projectile is the highest vertical position it reaches in its trajectory. Calculating the maximum height involves setting the final velocity at the peak (\(v\)) to zero, as the object momentarily stops ascending before descending.
Using the kinematic equation \(v^2 = v_0^2 + 2a (h - h_0)\), we can solve for the maximum height. In simpler terms, when the baseball is thrown upwards, it will continue rising until the upward movement is completely countered by gravity. In our problem, we calculate the highest point reached by the baseball, which turns out to be 156.25 ft. This knowledge is key to understanding projectile dynamics and motion limits under Earth's gravitational forces.