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Projectile motion describes the motion of an object thrown or projected into the air, subject to only the force of gravity. It's important to note that the force of gravity acts only in the vertical direction and it doesn't affect the horizontal motion of the object.

When analyzing our example of a ball thrown vertically upward, we are looking at a special case of projectile motion. Here, the horizontal component is null and only the vertical component is considered. As the ball rises, it slows down under the influence of gravity until it reaches its highest point, where the velocity is zero. Then, it starts its descent, accelerating in the downward direction until it reaches the original level. This type of motion is symmetric in the absence of air resistance, meaning that the time to rise to the highest point is equal to the time it takes to fall back to the starting height.

In physics, free fall acceleration refers to the acceleration experienced by an object due solely to the force of gravity. Commonly denoted as 'g', its value is approximately \( -9.8 \mathrm{m/s^2} \) on the surface of the Earth. The negative sign represents acceleration in the direction opposite to the initial velocity.

In our ball-throwing example, the ball experiences free fall acceleration once it leaves the thrower's hand. This acceleration affects its vertical motion throughout the journey. The ball's upward motion decelerates until the velocity becomes zero at the peak, and then the ball accelerates in the downward motion until it reaches the ground.

Displacement is a vector quantity which refers to the change in position of an object. In the case of the vertically thrown ball, the displacement corresponds to the height (altitude) it reaches above the level from which it was thrown. Velocity is also a vector and includes both the speed of the object and the direction of its motion. At the peak of its flight, the ball's velocity is zero because it is neither rising nor falling at that instant.

The initial velocity of the ball in our problem is \( 25 \mathrm{m/s} \) upward, and the ball's velocity when it returns to the ground is \( -25 \mathrm{m/s} \), which has the same magnitude but opposite direction, indicating downward motion. It's essential to understand the displacement and velocity in projectile motion since they are directly related to the kinematic calculations.

Kinematic equations allow us to calculate the motion of objects in kinematics, assuming constant acceleration. These equations relate displacement (\( s \)), initial velocity (\( u \)), final velocity (\( v \)), acceleration (\( a \)), and time (\( t \)).

For our example, the first equation we use is \( v^2 = u^2 + 2*a*s \), it helps to find the maximum height when the final velocity is zero. The second equation, \( v = u + at \) helps to determine the time taken to reach the highest point or return to the ground. Notably, kinematic equations are powerful tools in solving projectile motion problems, and understanding how to manipulate and use these equations is crucial for students studying physics.