Ch 2 – Motion In 1-D
2.1 Displacement, Velocity and Speed
Q: What does it really mean to say you are looking at motion in only 1 dimension?
Q: What does kinematics mean?
DEF: Displacement = the straight-line distance indicating an overall change in position of
an object.
x2 – x1
∆x
“as the crow flies”
DEF: Average velocity = the average rate of change of the displacement
Both displacement and average velocity can be either positive or negative.
Let’s do a couple of examples. Whenever you go through them in the text, make sure to
read the Remarks immediately following each example. There is often great information
regarding the set-up / analysis of the problem!
P 18, Ex 2-1
A comet moving toward the sun is first seen at 3.0 x 1012 m, relative to the sun. Exactly
one year later, it is seen at 2.1 x 1012 m. Find its displacement and average velocity. Be
mindful of resultant signs.
Ex Morr1
What would your displacement be if you ran exactly half-way
around a circular track with a circumference of 300 meters?
What would your average velocity be if you took 11 seconds
to make the trip? Convert that to mi/hour. Is that fast?
P 19, Ex 2-3
You run 100 in 12 s then turn around and jog 50 m back toward the starting point in 30 s.
Calculate:
a) Your average speed
b) Your average velocity for the entire trip
P 20, Ex 2-4
Two trains 75 km apart approach each other on parallel tracks, each moving at 15 km/h.
A bird flies back and forth between the trains at 20 km/h until the trains pass each other.
How far does the bird fly?
Below is a graphical depiction of average velocity. In the text, P1 is shown connected to P2
to form a straight line. The slope of this line forms the hypotenuse of a right triangle, and
the sides are, respectively, ∆t and ∆x. The reason we are allowed to do this is because of
the relationship between velocity and displacement, not velocity and distance!
It stands to reason that, using the process depicted above, the more erratic the curve, the
more off you actually are from the actual velocity.
Q: How could you get a more accurate picture of what the velocity actually is as you
traverse the curve?
Because this is a curve, the farther out you go
from your starting point (the larger your ∆t), the
less true the match is between the average
velocity and the actual path. What would happen
if you made your ∆t’s smaller and smaller?
This is really our first intro into Calculus:
DEF: Instantaneous speed = the magnitude of the instantaneous velocity
TRY p 22, Ex 2-5 NOW!!! I will choose a “volunteer” to come up and explain what is
going on.
See p 22, Ex 2-6
The position of a stone dropped from a cliff is described approximately by x = 5t2, where x
is in meters measured downward from the original position at t = 0, and t in is seconds.
Find the velocity at any time t.
Relative Velocity
Q: What is a frame of reference?
Mid-air refueling is a great example of a frame of
reference. Both planes are moving quite fast with
respect to the ground, but with respect to each
other, they are almost stationary!
Remember the lab we did in 10th grade pertaining to frame of reference with the two rollover buggies?
If an object moves with a velocity vpa relative to reference frame A, which is in turn
moving with velocity vab relative to reference frame B, the overall velocity is:
Vpb = vpa + vAB
EX: swimming with current, against current, running forward on a train that is moving
forward, backward, et cetera.
Really, that equation is now modified to accommodate for itty bitty particles traveling at
high speeds to be:
v pB =
v pA + v AB
1 + v pA v AB / c 2
where c is the speed of light.
This equation is not necessary for larger particles as the difference due to frames of
reference is negligible. For high-speed particles such as electrons, or light from distant
galaxies, the insertion of “c” becomes significant.
2.2 Acceleration
DEF: Acceleration = the rate of change of the instantaneous velocity.
DEF: Instantaneous acceleration is the limit of the ratio of
∆v
as ∆t → 0.
∆t
Q: On a plot of velocity versus time, what is the slope of the line tangent to the curve at a
specific time t?
A: The instantaneous acceleration, of course!
So, acceleration is the derivative of velocity with respect to time (dv/dt), and velocity the
derivative of displacement with respect to time (dx/dt).
Q: How would you depict acceleration with respect to displacement using the same
notation?
A: a = d2x
dt2
Try pp 25, 26, Ex’s 2-7 and 2-8. Also look at the Exercise in Dimensional Analysis on p 26.
Do Ch 2 HW Day 2. ☺