If you know the initial velocity \(v_{0}\) and the initial and final heights \(y_{0}\) and \(y,\) you can use Equation 2.10 to solve for the time \(t\) when the object will be at height \(y .\) But the equation is quadratic in \(t\) so you'll get two answers. Physically, why is this?

Kinematics is a branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that caused the motion. In dealing with kinematics, we focus on the trajectory and aspects of motion like velocity and acceleration, but we don't concern ourselves with what makes objects move.

When describing motion, we use various equations to calculate different parameters such as displacement, velocity, and time. For example, the kinematic equation for uniform acceleration is often written as: \( y = y_0 + v_0t - \frac{1}{2}gt^2 \), where \( y \) is the final position, \( y_0 \) the initial position, \( v_0 \) the initial velocity, \( t \) the time, and \( g \) the acceleration due to gravity. The negative sign indicates that gravity is acting in the opposite direction to the initial velocity.

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. It is a form of motion in which an object moves in a bilaterally symmetrical, parabolic path. The path that the projectile follows is determined by the initial speed (velocity), its initial angle of launch, and the gravitational force acting upon it.

In a typical projectile motion problem, you'll be given initial conditions and asked to find other parameters, like the height of the projectile at different times, the total time it is in the air, or the distance it travels. Remember, in projectile motion, horizontal and vertical motions are independent of each other, except they are tied together by the time.

Free fall is a special type of projectile motion where the initial horizontal velocity of the object is zero; it is motion under the influence of gravitational force only. Gravity is the only force acting on the object and it accelerates due to gravity uniformly. Any object that is being acted upon only by the force of gravity is said to be in a state of free fall. There are two important motions to note in free fall – the upward and downward movement.

Why Does an Object in Free Fall Accelerate?

In the absence of other forces, gravity causes the object to accelerate at a constant rate, adding to its velocity by approximately \(9.81 m/s^2\) each second on Earth. That's why when you drop an object, it doesn't just fall leisurely to the ground - it picks up speed rapidly.

We frequently talk about acceleration due to gravity, which is the acceleration that the Earth imparts to objects due to its gravitational pull. Near Earth's surface, this acceleration has a magnitude of approximately \(9.81 m/s^2\) and is directed downward towards the center of the Earth. It's an important value in physics because it's the rate at which all objects - irrespective of their mass - accelerate towards the Earth when dropped from the same height in a vacuum.

Universality of Acceleration Due to Gravity

It's fascinating that the acceleration due to gravity is constant for all objects (not taking into account air resistance and assuming a uniform gravitational field). This implies that if you drop a feather and a hammer from the same height, without air resistance, they would both hit the ground at the same time.