Two balls are pushed off a table of height \(h\). Ball \(a\) has negligible horizontal velocity and falls straight down. Ball \(b\) has an initial horizontal velocity \(v_{o x}\). Neglecting air resistance: (A) ball \(a\) hits the ground first. (B) ball \(b\) hits the ground first. (C) they both hit the ground at the same time, but \(a\) 's speed is greater (D) they both hit the ground at the same time, but \(b\) 's speed is greater. (E) they both hit the ground at the same time with the same speed.

Understanding projectile motion is crucial for solving a variety of physics problems. It is the motion experienced by an object that is thrown near the Earth's surface and moves along a curved path under the action of gravity alone. This motion can be understood as two separate motions: horizontal motion without any acceleration (since no forces act horizontally in our case) and vertical motion with constant acceleration due to gravity.

When an object like ball (b) is given an initial horizontal velocity (\(v_{ox}\)) and is subject to gravity, its horizontal motion continues at a constant velocity since there is no air resistance or other forces to alter that motion. Its vertical motion, on the other hand, behaves like free fall.
For our exercise, understanding that the horizontal and vertical components of motion are independent allows us to conclude that the presence of an initial horizontal velocity does not affect the time it takes for an object to hit the ground. Therefore, both balls hit the ground simultaneously, regardless of any initial horizontal velocity that ball (b) may have.

Kinematic equations play a key role in predicting the future position and velocity of an object moving under a constant acceleration without the need for understanding the forces causing the acceleration. In the context of our problem, we used the equation \(h = v_{oy}t + \frac{1}{2}gt^2\) to determine the time it takes for the balls to hit the ground.

The variables in play are the initial velocity in the y-direction (\(v_{oy}\)), which is zero for both balls, the gravitational acceleration (\(g\)), and the height from which the balls fall (\(h\)). Another kinematic equation used is \(v_{y} = v_{oy} + gt\), giving us the final velocity in the vertical direction when the balls hit the ground.
By applying these equations correctly, we can determine the timing of events and the final velocities in both the vertical and horizontal directions. Since the horizontal and vertical motions are decoupled, the balls will hit the ground at the same time irrespective of their horizontal velocities.

Free fall is a particular type of projectile motion where an object is only subjected to the force of gravity and accelerates downwards. It's assumed that there are no other forces acting, such as air resistance.

In our exercise, both balls are in free fall once they leave the edge of the table. The force of gravity will cause them to accelerate towards the ground at the same rate of \(9.8 \text{m/s}^2\), which is the standard acceleration due to gravity near the Earth's surface. As they're only being acted on by gravity, it is the only force influencing their motion, thus they share the same falling time to the ground.
It's very important to remember that in free fall, the horizontal motion of an object doesn't affect the vertical acceleration due to gravity. Therefore, ball (a), with negligible horizontal motion, and ball (b), with an initial horizontal velocity, will hit the ground at the same time if dropped from the same height.