- Initial position and velocity are given a subscript 0; final values have no subscript. Thus,
Δt = t
Δx = x−x 0
Δv = v−v 0
⎫
⎭
⎬
• The following kinematic equations for motion with constantaare useful:
x=x 0 +v
-
t
v-=
v 0 +v
2
v=v 0 +at
x=x 0 +v 0 t+^1
2
at^2
v^2 =v 02 + 2a(x−x 0 )
• In vertical motion,yis substituted forx.
2.6 Problem-Solving Basics for One-Dimensional Kinematics
- 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?
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 gravityg, which averages
g= 9.80 m/s
2
.
• Whether the accelerationashould be taken as+gor−gis determined by your choice of coordinate system. If you choose the upward
direction as positive,a= −g= −9.80 m/s^2 is negative. In the opposite case,a= +g = 9.80 m/s^2 is positive. Since acceleration is
constant, the kinematic equations above can be applied with the appropriate+gor−gsubstituted fora.
- 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 displacementxvs. timetis velocityv.
• The slope of a graph of velocityvvs. timetgraph is accelerationa.
- Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.
Conceptual Questions
2.1 Displacement
1.Give an example in which there are clear distinctions among distance traveled, displacement, and magnitude of displacement. Specifically identify
each quantity in your example.
2.Under what circumstances does distance traveled equal magnitude of displacement? What is the only case in which magnitude of displacement
and displacement are exactly the same?
3.Bacteria move back and forth by using their flagella (structures that look like little tails). Speeds of up to50 μm/s⎛⎝50×10−6m/s⎞⎠have been
observed. The total distance traveled by a bacterium is large for its size, while its displacement is small. Why is this?
2.2 Vectors, Scalars, and Coordinate Systems
4.A student writes, “A bird that is diving for prey has a speed of −10 m/s.” What is wrong with the student’s statement? What has the student
actually described? Explain.
5.What is the speed of the bird inExercise 2.4?
6.Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
7.A weather forecast states that the temperature is predicted to be−5ºCthe following day. Is this temperature a vector or a scalar quantity?
Explain.
2.3 Time, Velocity, and Speed
8.Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.
CHAPTER 2 | KINEMATICS 77