Moving from graphs to equations
a = dv/dt, where v = dx/dt. And a is a constant.
We get four 1-D (constant-acceleration) equations from
these facts. (The book lists three equations, but there’s a
challenge in Before Class 5: Can you derive the other?
And what if acceleration were not constant? So long as we
know how a varies with time, a(t), we can derive a useful
set of equations. But the equations with constant acceleration are especially significant—because we live with it (g).
10/10/14
Oregon State University PH 211, Class #6
1
The Translational Kinematic Equations
vf = vi + at
(doesn’t use x)
x = vit + (1/2)a(t)2
(doesn’t use vf)
vf2 = vi2 + 2ax
(doesn’t use t)
x = (1/2)(vi + vf)t
(doesn’t use a)
Five variables (x, vi, vf, a and t): If you know any
three, you can usually solve for the other two.
Note: All variables except t are vector quantities;
their signs (±) indicate their directions.
10/10/14
Oregon State University PH 211, Class #6
2
So: You look at your watch (that’s time ti) when the
object is at position xi along your tape measure; you
look at the watch again (at time tf) when the object
is at position xf. (Then in the equations, you use
x = xf – xi and t = tf – ti)
Caution: These equations apply to the motion of a
rigid body only when a is constant. The object may
have only one value for its acceleration vector
throughout the entire time interval t. Whenever the
value of a changes, you must start a fresh analysis.
10/10/14
Oregon State University PH 211, Class #6
3
The kinematics equations offer a tool like a picture
frame that you set over all or part of a situation to
compute an object’s motion.
Example: You’re driving onto the freeway,
accelerating ahead, and you suddenly see traffic
stopped ahead. You move your foot to the brake
and apply it steadily.
What three applications of the kinematics equations
must you make here?
10/10/14
Oregon State University PH 211, Class #6
4
A ball is tossed vertically upward from a bridge at
an initial speed of 29.4 m/s. Defining upward as
the positive y-direction, find its velocity and
position 8.00 seconds later. (Use earth’s local freefall acceleration magnitude: g = 9.80 m/s2)
1.
vf = 49.0 m/s; y = –78.4 m
2.
vf = –49.0 m/s;
y = –78.4 m
3.
vf = –78.4 m/s;
y = –314 m
4.
vf = –49.0 m/s; y = –167 m
5.
None of the above.
10/10/14
Oregon State University PH 211, Class #6
5
If an object is accelerating toward a point,
then it must be getting closer and closer to
that point.
1. True
2. False
3. Not enough information
(“it depends”)
10/10/14
Oregon State University PH 211, Class #6
6
Standing in the middle of the MU quad,
you throw a ball straight up in the air.
At the ball’s highest point,…
A.
B.
C.
D.
E.
10/10/14
vball = 0
and
aball = 0
vball ≠ 0
and
aball = 0
vball = 0
and
aball ≠ 0
vball ≠ 0
and
aball ≠ 0
Not enough information
(“it depends”).
Oregon State University PH 211, Class #6
7
Ball A is dropped from rest from a tall building.
2 seconds later, ball B is dropped from rest from
the same point. Once both balls are in motion:
• Does their velocity difference increase,
decrease, or remain the same with time?
• Do they get further apart, closer together, or stay
the same distance apart?
Ignore air drag and assume a constant value of g
(local free-fall acceleration). Solve this with
time graphs—no calculations!
10/10/14
Oregon State University PH 211, Class #6
8
A ball is thrown straight upward with an initial
speed of vi. Assume no air drag and a constant
free-fall acceleration, g.
• How high does the ball go (release to peak)?
• How long does it take to reach that peak?
• How long does it take for the round trip (return
to its release point)?
• What is the ball’s velocity when it returns to its
release point?
10/10/14
Oregon State University PH 211, Class #6
9
From a bridge high above a river, ball A is thrown
straight up with initial speed |vi|. Ball B is thrown
straight down with the same initial speed, |vi|.
Each hits the water. Compare their impact speeds.
1.
2.
3.
4.
|vimpact.A| > |vimpact.B|
|vimpact.A| < |vimpact.B|
|vimpact.A| = |vimpact.B|
There is not enough information (“it depends”).
10/10/14
Oregon State University PH 211, Class #6
10
A rocket lifts off (from rest) from the earth.
During its boost phase, it has a vertically upward
constant acceleration value aboost.
At a time tb after lift-off, a bolt falls from the side
of the rocket. (Probably not good.)
Assuming no wind or air drag and a constant freefall g value, draw the time graphs (a-t, v-t and y-t)
for the bolt, from lift-off to impact.
10/10/14
Oregon State University PH 211, Class #6
11
This a-t graph describes a particle moving along
a horizontal axis (+ direction is to the right).
• The particle’s initial velocity (vi) is -5 m/s, and
its initial position (xi) is -20 m.
• Find its position and velocity at t = 10 s.
10/10/14
Oregon State University PH 211, Class #6
12
Not all motions have constant acceleration!
Before its engine gets up to full power (that
is, for its first 5 seconds of operation), the
acceleration of a rocket sled is described by
a(t) = 3t2 + 1 (m/s2)
How far does the sled travel, starting from
rest, in those first 5 seconds?
10/10/14
Oregon State University PH 211, Class #6
13