After a particle moves along the x axis from some initial position xi to some final position xf, its displacement is
The average velocity of a particle during some time interval is the displacement delta x divided by the time interval delta t during which that displacement occurred:
The average speed of a particle is equal to the ratio of the total distance it travels to the total time it takes to travel that distance.
The instantaneous velocity of a particle is defined as the limit of the ratio delta x /delta tdelta t as approaches zero. By definition, this limit equals the derivative of x with respect to t, or the time rate of change of the position:
The instantaneous speed of a particle is equal to the magnitude of its velocity. The average acceleration of a particle is defined as the ratio of the change in its velocity delta vx divided by the time interval delta t during which that change occurred:
The instantaneous acceleration is equal to the limit of the ratio delta vx / delta t as delta t approaches 0. By definition, this limit equals the derivative of vx with respect to t, or the time rate of change of the velocity:
The equations of kinematics for a particle moving along the x axis with uniform acceleration ax (constant in magnitude and direction) are
You should be able to use these equations and the definitions in this chapter to analyze the motion of any object moving with constant acceleration.
An object falling freely in the presence of the Earth’s gravity experiences a free-fall acceleration directed toward the center of the Earth. If air resistance is neglected, if the motion occurs near the surface of the Earth, and if the range of the motion is small compared with the Earth’s radius, then the free-fall acceleration g is constant over the range of motion, where g is equal to 9.80 m/s2.
Complicated problems are best approached in an organized manner. You should be able to recall and apply the steps of the GOAL strategy when you need them.