Sections

Section Summary
# Section 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 $\mathrm{\Delta}x$ is defined to be
$$\mathrm{\Delta}x={x}_{\mathrm{f}}-{x}_{0},$$where ${x}_{0}$ is the initial position and ${x}_{\mathrm{f}}$ is the final position. In this text, the Greek letter $\mathrm{\Delta}$ (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.

### 2.2 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.

### 2.3 Time, Velocity, and Speed

- Time is measured in terms of change, and its SI unit is the second (s). Elapsed time for an event is
$$\mathrm{\Delta}t={t}_{\mathrm{f}}-{t}_{0},$$where ${t}_{\mathrm{f}}$ 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 $t\text{.}$
- Average velocity $\stackrel{-}{v}$ is defined as displacement divided by the travel time. In symbols, average velocity is
$$\stackrel{-}{v}=\frac{\mathrm{\Delta}x}{\mathrm{\Delta}t}=\frac{{x}_{\text{f}}-{x}_{0}}{{t}_{\text{f}}-{t}_{0}}\text{.}$$
- The SI unit for velocity is 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.

### 2.4 Acceleration

- Acceleration is the rate at which velocity changes. In symbols, average acceleration $\stackrel{-}{a}$ is
$$\stackrel{-}{a}=\frac{\mathrm{\Delta}v}{\mathrm{\Delta}t}=\frac{{v}_{\mathrm{f}}-{v}_{0}}{{t}_{\mathrm{f}}-{t}_{0}}\text{.}$$
- The SI unit for acceleration is ${\text{m/s}}^{2}\text{.}$
- 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.

### 2.5 Motion Equations for Constant Acceleration in One Dimension

- To simplify calculations we take acceleration to be constant, so that $\stackrel{-}{a}=a$ 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,
$$\left.\begin{array}{lll}\mathrm{\Delta}t& =& \mathrm{t.}\\ \mathrm{\Delta}x& =& x-{x}_{0.}\\ \mathrm{\Delta}v& =& v-{v}_{0.}\end{array}\right\}$$
- The following kinematic equations for motion with constant $a$ are useful
$$x={x}_{0}+\stackrel{-}{v}\mathrm{t,}$$$$\stackrel{-}{v}=\frac{{v}_{0}+v}{2}\text{,}$$$$v={v}_{0}+\text{at,}$$$$x={x}_{0}+{v}_{0}t+\frac{1}{2}{\text{at}}^{2}\text{,}$$$${v}^{2}={v}_{0}^{2}+2a\left(x-{x}_{0}\right)\text{.}$$
- In vertical motion, $y$ is substituted for $x\text{.}$

### 2.6 Problem-Solving Basics for One-Dimensional Kinematics

*The six basic problem-solving steps for physics are as follows:**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?

### 2.7 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 $g\text{,}$ which averages
$$g=9\text{.}{\text{80 m/s}}^{2}.$$
- 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, $a=-g=-9\text{.}\text{80 m}{\text{/s}}^{2}$ is negative. In the opposite case, $a=\mathrm{+g}=9\text{.}{\text{80 m/s}}^{2}$ is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate $+g$ or $-g$ substituted for $a\text{.}$ - For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.

### 2.8 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 $v\text{.}$
- The slope of a graph of velocity $v$ vs. time $t$ graph is acceleration $a\text{.}$
- Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.