Chapter Summary

2.1 Displacement

Kinematics is the study of motion without considering its causes. In this chapter, it is limited to motion along a straight line, called one-dimensional motion.
Displacement is the change in position of an object.
In symbols, displacement [latex]Δ x[/latex] is defined to be
[latex]Δ x = x_{f} - x_{0} ,[/latex]

where [latex]x_{0}[/latex] is the initial position and [latex]x_{f}[/latex] is the final position. In this text, the Greek letter [latex]Δ[/latex] (delta) always means “change in” whatever quantity follows it. The SI unit for displacement is the meter (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.

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.

Time is measured in terms of change, and its SI unit is the second (s). Elapsed time for an event is
[latex]Δ t = t_{f} - t_{0} ,[/latex]

where [latex]t_{f}[/latex] is the final time and [latex]t_{0}[/latex] 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 [latex]t[/latex].
Average velocity [latex]\overset{-}{v}[/latex] is defined as displacement divided by the travel time. In symbols, average velocity is
[latex]\overset{-}{v} = \frac{Δ x}{Δ t} = \frac{x_{\text{f}} - x_{0}}{t_{\text{f}} - t_{0}} .[/latex]
The SI unit for velocity is m/s.
Velocity is a vector and thus has a direction.
Instantaneous velocity [latex]v[/latex] 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.

Acceleration is the rate at which velocity changes. In symbols, average acceleration [latex]\overset{-}{a}[/latex] is
[latex]\overset{-}{a} = \frac{Δ v}{Δ t} = \frac{v_{f} - v_{0}}{t_{f} - t_{0}} .[/latex]
The SI unit for acceleration is [latex]\text{m}/\text{s}^{2}[/latex].
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 [latex]a[/latex] is the acceleration at a specific instant in time.
Deceleration is an acceleration with a direction opposite to that of the velocity.

To simplify calculations we take acceleration to be constant, so that [latex]\overset{-}{a} = a[/latex] at all times.
We also take initial time to be zero.
Initial position and velocity are given a subscript 0; final values have no subscript. Thus,
[latex]\left\begin{matrix} Δ t & = & t \\ Δ x & = & x - x_{0} \\ Δ v & = & v - v_{0} \end{matrix}\right}[/latex]
The following kinematic equations for motion with constant [latex]a[/latex] are useful:
[latex]x = x_{0} + \overset{-}{v} t[/latex]

[latex]\overset{-}{v} = \frac{v_{0} + v}{2}[/latex]

[latex]v = v_{0} + \text{at}[/latex]

[latex]x = x_{0} + v_{0} t + \frac{1}{2} \text{at}^{2}[/latex]

[latex]v^{2} = v_{0}^{2} + 2 a \left(x - x_{0}\right)[/latex]
In vertical motion, [latex]y[/latex] is substituted for [latex]x[/latex].

The six basic problem solving steps for physics are:

Step 1. Examine the situation to determine which physical principles are involved.
Step 2. Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
Step 3. Identify exactly what needs to be determined in the problem (identify the unknowns).
Step 4. Find an equation or set of equations that can help you solve the problem.
Step 5. Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units.
Step 6. Check the answer to see if it is reasonable: Does it make sense?

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 [latex]g[/latex], which averages
[latex]g = 9 . \text{80 m}/\text{s}^{2} .[/latex]
Whether the acceleration a should be taken as [latex]+ g[/latex] or [latex]- g[/latex] is determined by your choice of coordinate system. If you choose the upward direction as positive, [latex]a = - g = - 9 . \text{80 m} /\text{s}^{2}[/latex] is negative. In the opposite case, [latex]a = +g = 9 . \text{80 m}/\text{s}^{2}[/latex] is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate [latex]+ g[/latex] or
[latex]- g[/latex] substituted for [latex]a[/latex].
For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.

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 [latex]x[/latex] vs. time [latex]t[/latex] is velocity [latex]v[/latex].
The slope of a graph of velocity [latex]v[/latex] vs. time [latex]t[/latex] graph is acceleration [latex]a[/latex].
Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.

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