CH 3: 3-3 Position, Velocity and Acceleration
DEF: Position vector is drawn from the origin, or what you are calling the
starting point, to the position of the particle. The actual path is given by the
displacement vector.
NOTE: The biggest difference that distinguishes this chapter from last (and
from what you know from Calculus) is that these graphs are often y / x graphs,
NOT x / t (or v / t or a / t), like you are used to seeing.
The notation for each is given by:
Consider the axes carefully. What are you actually looking at in the picture
below? What does each vector represent? How do you know?
r1 ________________ r2 = _ ________________ ∆r = ______________
Q What is the ratio of the displacement vector to the time interval? Why is
that true?
A: It is the average velocity. It is over the entire interval, so it’s the average
for the interval.
__
Q: When is the magnitude of the displacement vector less than the distance
traveled? When is it equal to the distance traveled?
A: When the path is a curve, and the displacement vector “cuts across” the
curve, like in the diagram above. When the path is a straight line, or when the
time intervals shrink to zero.
Q: How would describe the direction of the displacement vector as ∆t goes to
zero? What would you call it?
A: It approaches the tangent to the curve. It is the instantaneous velocity.
This instantaneous velocity vector’s magnitude is its speed and its direction is
the direction of the particle along the line tangent to the curve.
So, v = dx/dt i + dy/dt j = vx i + vy j
TRY EX 3-3, P 60
Q: Why not just take the slope of the graph to get the instantaneous velocity?
A: This graph is y with respect to x, not x with respect to t.
Note: When expressing your answer, you need to be mindful of:
1. Sig figs
2. The different ways you may be asked to present.
• This answer may be modified to be 0.2 m/s at – 300 SE, (or 3000),
depending on how you are asked to report.
Relative Velocity
Q: What does relative velocity mean? How do you find it?
A: Velocity with respect to a certain frame of reference. IN order to calculate
the overall relative velocity, you have to find each part, with respect to its
reference frame, and then take into account the cumulative effect of all of the
reference frames.
Just like before, if you want to find total velocity (or any other vector, for that
matter), you can add or subtract the vectors using any of the three methods for
vector addition.
SEE EX 3-4, P 61-62
Q: Why is the vector for velocity of plane wrt air oriented as such? What does
he horizontal vector describe? What does the vertical vector describe?
A: Just like with the car pulling another car across the paper lab you did in
10th grade, if you want to get the plane to fly north, but there is an easterly
wind, you have to fly it to the west to accommodate. The horizontal vector is
the velocity of the air wrt ground and the vertical vector is the resultant velocity.
The Acceleration Vector
Average acceleration and instantaneous acceleration are given by the
respective equations below.
Q: If a car is traveling at a constant speed around a corner, what is happening
to its acceleration? Why?
A: Its acceleration is changing as the direction is constantly changing.
SEE EX 3-6 P 63
For diagrams (a) and (b) below, the vectors represent the acceleration of a
bungee jumper slowing down just before reversing direction. The faster she
moves, the grater the distance between the marks, and vice-versa. The average
acceleration from t2 to t4 = ∆v / ∆t, where ∆v = v4 – v2 and ∆t = t4 – t2. This
would let you estimate her acceleration at time t3. Since a3 and ∆v are in the
same direction, by finding the direction of ∆v, we can also find the direction of
a3. We get the direction of ∆v by using v2 + ∆v = v4 to draw what we see in
diagram (b).
NOTE: Because the jumper is moving faster at time t2 than at t4, v2 is longer
than v4. It’s this information that gives us the direction of ∆v, and thus the
direction of a3.
SEE EX 3-7 P 64
NOTE: Finding the direction of a using a motion diagram is not exact, so it’s
only an estimation for the direction of a.