Summary page

Below is a summary of this chapter's main concepts, organized per theory page, and starting with Displacement.

Summary of Displacement

We start by studying motion without considering its causes, which is called kinematics. In this chapter, we limited the study to motion along a straight line, called one-dimensional motion.
Displacement is the change in position of an object.
In symbols, displacement #\Delta x# is defined to be \[ \Delta x={x}_{f}-{x}_{0}\] where #{x}_{0}# is the initial position and #{x}_{f}# is the final position. In this text, the Greek letter #\Delta # (delta) always means “change in” whatever quantity follows it. The SI unit for displacement is the meter (#\unit{\!}{m}#). Displacement has a direction as well as a magnitude.
When you start a problem, assign which direction will be positive.
Distance is the magnitude of displacement between two positions.
Distance traveled is the total length of the path traveled between two positions.

Next, we learned the distinction between vectors and scalars in Vectors, scalars, and coordinate systems.

Summary of Vectors, scalars, and coordinate Systems

A vector is any quantity that has magnitude and direction.
A scalar is any quantity that has magnitude but no direction.
Displacement and velocity are vectors, whereas distance and speed are scalars.
In one-dimensional motion, direction is specified by a plus or minus sign to signify left or right, up or down, and the like.

This allowed us to distinguish between the notions of velocity and speed in Time, velocity, and speed.

Summary of Time, velocity, and speed

Time is measured in terms of change, and its SI unit is the second (#\unit{\!}{s}#). Elapsed time for an event is \[ \Delta t={t}_{f}-{t}_{0}\] where #{t}_{f}# is the final time and #{t}_{0}# is the initial time. The initial time is often taken to be zero, as if measured with a stopwatch; the elapsed time is then just #t#.
Average velocity #\bar{v}# is defined as displacement divided by the travel time. In symbols, average velocity is \[ \bar{v}=\dfrac{\Delta x}{\Delta t}=\dfrac{{x}_{f}-{x}_{0}}{{t}_{f}-{t}_{0}}\]
The SI unit for velocity is the meter per second #\unit{\!}{m/s}#.
Velocity is a vector and thus has a direction.
Instantaneous velocity #v# is the velocity at a specific instant or the average velocity for an infinitesimal interval.
Instantaneous speed is the magnitude of the instantaneous velocity.
Instantaneous speed is a scalar quantity, as it has no direction specified.
Average speed is the total distance traveled divided by the elapsed time (average speed is not the magnitude of the average velocity). Speed is a scalar quantity; it has no direction associated with it.

We continued our study of motion in one dimension in Acceleration.

Summary of Acceleration

Acceleration is the rate at which velocity changes. In symbols, average acceleration #\bar{a}# is \[ \bar{a}=\dfrac{\Delta v}{\Delta t}=\dfrac{{v}_{f}-{v}_{0}}{{t}_{f}-{t}_{0}}\]
The SI unit for acceleration is the meter per second squared #\unit{\!}{m/s^2}#.
Acceleration is a vector, and thus has a both a magnitude and direction.
Acceleration can be caused by either a change in the magnitude or the direction of the velocity.
Instantaneous acceleration #a# is the acceleration at a specific instant in time.
Deceleration is an acceleration with a direction opposite to that of the velocity.

With the basic concepts established, we then learned which equations describe motion in Motion equations for constant acceleration in one dimension.

Summary of Motion equations for constant acceleration in one dimension

To simplify calculations, we always assume a constant acceleration, so that #\bar{a}=a# at all times.
We also take the initial time to be zero, that is, #t_0=0#, which implies #t_f=t# and #\Delta t=t#.
Initial position and velocity are given a subscript #0#; final values have no subscript. Thus,
\[\begin{array}{rcl}\Delta t&=&t\\ \Delta x&=&x-{x}_{0}\\ \Delta v&=&v-{v}_{0}\end{array}\]
The following equations for motion with constant #a# are useful \[ x={x}_{0}+\bar{v}\cdot t\]\[ \bar{v}=\dfrac{{v}_{0}+v}{2}\]\[ v={v}_{0}+a\cdot t\]\[ x={x}_{0}+{v}_{0}\cdot t+\dfrac{1}{2}\cdot a\cdot t^2\]\[ {v}^{2}={v}_{0}^{2}+2\cdot a\cdot \left(x-{x}_{0}\right)\]
In vertical motion, #y# is substituted for #x#.

We continued with strategies for solving problems in Problem-solving basics for one-dimensional motion.

Summary of Problem-solving basics for one-dimensional kinematics

Summary of Graphical analysis of one-dimensional motion

Graphs of motion can be used to analyze motion.
Graphical solutions yield identical solutions to mathematical methods for deriving motion equations.
The slope of a graph of displacement #x# vs. time #t# is velocity #v#.
The slope of a graph of velocity #v# vs. time #t# graph is acceleration #a#.
Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.

Finally, we focused our equations to cases of objects affected only by gravity in Falling objects.

Summary of Falling Objects

An object in free fall experiences constant acceleration if air resistance is negligible.
On Earth, all free-falling objects have an acceleration due to gravity #g#, which averages \[ g=\unit{9.81}{m/s^2}\]
Whether the acceleration #a# should be taken as #+g# or #-g# is determined by your choice of coordinate system. If you choose the upward direction as positive, #a=-g=-\unit{9.81}{m/s^2}# is negative. In the opposite case, #a=+g=\unit{9.81}{m/s^2}# is positive. Since acceleration is constant, the equations of motion above can be applied with the appropriate #+g# or #-g# substituted for #a#.
For objects in free fall, upwards is usually taken as positive for displacement, velocity, and acceleration.