# Kinematics

## Falling Objects


### Learning Objectives

By the end of this section, you will be able to:
1. Describe the effects of gravity on objects in motion.
2. Describe the motion of objects that are in free fall.
3. Calculate the position and velocity of objects in free fall.
Falling objects form an interesting class of motion problems. For example, we can estimate the depth of a vertical mine shaft by dropping a rock into it and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, we can examine some interesting situations and learn much about gravity in the process.

### Gravity

The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the same constant acceleration, independent of their mass. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones. 
In the real world, air resistance can cause a lighter object to fall slower than a heavier object of the same size. A tennis ball will reach the ground after a hard baseball dropped at the same time. (It might be difficult to observe the difference if the height is not large.) Air resistance opposes the motion of an object through the air, while friction between objects—such as between clothes and a laundry chute or between a stone and a pool into which it is dropped—also opposes motion between them. For the ideal situations of these first few chapters, an object falling without air resistance or friction is defined to be in free-fall.
The force of gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called the acceleration due to gravity. The acceleration due to gravity is constant, which means we can apply the kinematics equations to any falling object where air resistance and friction are negligible. This opens a broad class of interesting situations to us. The acceleration due to gravity is so important that its magnitude is given its own symbol, . It is constant at any given location on Earth and has the average value
Although  varies from  to , depending on latitude, altitude, underlying geological formations, and local topography, the average value of  will be used in this text unless otherwise specified. The direction of the acceleration due to gravity is downward (towards the center of Earth). In fact, its direction defines what we call vertical. Note that whether the acceleration  in the kinematic equations has the value  or  depends on how we define our coordinate system. If we define the upward direction as positive, then , and if we define the downward direction as positive, then .

### One-Dimensional Motion Involving Gravity

The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So we start by considering straight up and down motion with no air resistance or friction. These assumptions mean that the velocity (if there is any) is vertical. If the object is dropped, we know the initial velocity is zero. Once the object has left contact with whatever held or threw it, the object is in free-fall. Under these circumstances, the motion is one-dimensional and has constant acceleration of magnitude . We will also represent vertical displacement with the symbol  and use  for horizontal displacement.

### Test Prep for AP Courses


### Section Summary

1. An object in free-fall experiences constant acceleration if air resistance is negligible.
2. On Earth, all free-falling objects have an acceleration due to gravity , which averages
    
3. Whether the acceleration a should be taken as  or  is determined by your choice of coordinate system. If you choose the upward direction as positive,  is negative. In the opposite case,  is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate  or
 substituted for .
4. For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.

### Conceptual Questions


### Problems & Exercises

Assume air resistance is negligible unless otherwise stated.