Using concepts and physical quantities discussed in this chapter, explain why it is usually safe for a person standing on the seat of a chair to jump horizontally and land on the floor, but not for a person standing on the roof of a tall building to jump horizontally and land on the ground.

In physics, free-fall motion refers to the movement of an object under the influence of gravitational force only, without any air resistance. It is important to remember that in free-fall, all objects accelerate at the same rate, regardless of their mass. This uniform acceleration is due to the pull of gravity.

When a person jumps from any height, they initially have a vertical velocity (\(v_{0y}\)) of zero. As they fall, their velocity increases because of the gravitational pull acting upon them. In the classroom example, when jumping from a chair, the distance of free-fall is relatively small, which keeps the acceleration phase short and results in a lower landing velocity and thus less impact force.

Free-fall motion is perfectly symmetrical, meaning that the time an object takes to reach its peak height is equal to the time it takes to fall back down from that height, assuming it lands at the same elevation from which it was launched. Knowing this assists in understanding the time of flight and impact velocities which are crucial for evaluating safety in falling scenarios.

The rate at which an object accelerates when in free-fall is known as gravitational acceleration, often represented by the symbol (\(g\)) and its value on Earth averages to approximately \(9.81 m/s^2\). Gravitational acceleration is crucial in calculating the time of flight and final velocities for free-falling objects.

Using the kinematic equations, one can determine the velocity of an object at any point during its fall. For an object released from rest, like our person jumping from a chair or building, the final vertical velocity (\(v_y\)) can be directly calculated from the height (\(h\)) they fall using the equation \(v_y^2 = 2gh\).

A larger height of fall results in a longer time under gravitational acceleration, leading to higher final velocities upon impact. This is a primary reason why falling from a greater height is more dangerous, and understanding gravitational acceleration is key to analyzing the risks involved.

The impact force upon landing is the force experienced by an object when it comes to a sudden stop after a fall. It's a critical factor to consider in safety evaluations of falling objects, including humans. The impact force increases with the square of the velocity at which the object impacts the ground because it's proportional to both the mass and the square of the velocity with which the person lands.The formula to calculate the force is derived from the impulse-momentum theorem, stating that the force multiplied by the time during which the force acts (impact time) is equal to the change in momentum. However, in practical terms, it is sometimes enough to know that a higher speed at impact will result in a greater force. Since the impact force is what causes injuries, one can infer that a fall from a greater height will cause greater injury or even be fatal due to the higher velocity acquired during free-fall and resulting in a stronger force of impact upon landing. Thus, it's advised that when engaging in any activity that involves jumping or falling from heights, the height should be low enough to ensure a safe landing, where the legs can absorb the shock without causing harm.