Vertical free-fall

Motion that occurs solely under the influence of gravity is called free-fall. This term applies as much to satellites orbiting the earth as to bodies moving vertically up or down. Even today, with stopwatches that can measure to 0.01 s, it is not easy to determine how the speed of a falling body varies. Thus, it is hardly surprising that the nature of this motion was unclear for centuries. In the early seventeenth century, Galileo established an im­portant fact:

In the absence of air resistance, all falling bodies have the same acceleration due to gravity, regardless of their sizes or shapes.

The value of the acceleration due to gravity depends on both latitude and altitude. It is approximately 9.8 m/s² near the surface of the earth. In the presence of air resistance, the acceleration of a falling body is only approximately constant. The acceleration diminishes with time and can even become zero. For low speeds and short time intervals one can neglect this complication, and assume that bodies are in free-fall with a constant acceleration. In this case, we may apply the equations of kinematics.

Because we will later use the x axis for horizontal motion, we use the у axis for vertical motion. If the у axis points upward, the acceleration due to gravity is a = -g j, where g = 9.8 m/s² (a positive scalar). With a_y = -g, the equations of kinematics now read

(2.11)

(2.12)

The quantities υ ₀ and υ are the у components of the vectors v ₀ and v. Their signs are determined by their directions relative to the chosen +y axis. Note in particu­lar that the sign of the acceleration does not depend on whether the body is going up or coming down. It is important to clearly indicate the origin so that a value may be assigned to the initial vertical position coordinate y₀.

FIGURE 2.11The horizontal motion of a projectile is at constant velocity while thevertical motion occurs at constant acceleration (provided air resistance is negligible). The vertical component of the motion of a ball projected horizontally is the same as that of a ball thatis simply dropped.

Дата добавления: 2014-01-05