Motion Diagrams
Skiing through the woods
Motion Graphs
What kind of motion is this?
What kind of motion is this?
What is the velocity during each
segment? Graph it!
∆x
v=
∆t
v2 = 0m / s
=
v1
400m
= 2m / s
200 s
−400m
v3 =
= −1m / s
400 s
Which position-versus-time graph
represents the motion shown in the
motion diagram?
Which position-versus-time graph
represents the motion shown in the
motion diagram?
Motion Graphs
What kind of motion is this?
Deriving Graphs from Graphs
x(t ) =
−4t + 2t 2
v (t ) =−4 + 4t
a = 4m / s
a x = 4m / s 2
2
QuickCheck 2.7
Here is a position graph
of an object:
At t = 3.0 s, the object’s
velocity is
A.
B.
C.
D.
E.
40 m/s.
20 m/s.
10 m/s.
–10 m/s.
None of the above.
Slide 2-50
QuickCheck 2.7
Here is a position graph
of an object:
At t = 3.0 s, the object’s
velocity is
A.
B.
C.
D.
E.
40 m/s.
20 m/s.
10 m/s.
–10 m/s.
None of the above.
Slide 2-51
Which velocity-versus-time graph or
graphs goes with this acceleration-versustime graph? The particle is initially moving
to the right and eventually to the left.
Which velocity-versus-time graph or
graphs goes with this acceleration-versustime graph? The particle is initially moving
to the right and eventually to the left.
Free Fall
Unless told otherwise, ignore air resistance for
free fall problems!
Falling from Rest
v=
v0 + gt
f
1 2
∆y = v0t + gt
2
Take
down
as +y:
+
Estimate :
a = g ~ 10m / s
2
=
v at= 10t
1 2
2
∆
=
y
at
= 5t
2
How FAR is not
How FAST!
v ≠ ∆y !
v0 = 0
v = 10m / s
∆y =5m
v = 20m / s
∆y =20m
v = 30m / s
∆y =45m
v = 40m / s
∆y =
80m
v = 50m / s
∆y =
125m
How Far: y(t) ~ t2
1 2
∆y= v0 t + at
2
+
How Fast: v(t) ~ t1
v=
v0 + at
f
How Fast How Fast is
Changing!
g = 9.80m / s
2
FIRST: Define Reference Frame
In this reference frame,what is the sign of a?
a = −9.80m / s
2
What is v at t = 3s?
Knowns : v0 =
0, a =
−9.8m / s 2 , t =
3s
Unknown : v f = ?
v=
v0 + at
f
m
= 0 − 9.80 2 (3s )
s
m
= −29.4
s
Negative because it is moving downward, in the negative direction!
FIRST: Define Reference Frame
How far did the ball fall in those 3 seconds?
Knowns : v0 =
0, a =
−9.8m / s 2 , t =
3s, v f =
−29.4m / s
Unknown : ∆y =
?
1
∆y = v0t + at
2
2
m
2
=0 + (−9.8 2 )(3s )
s
2
1
= −44.1m
The ball fell 44.1m.
The displacement is negative because it is moves downward, in the negative
direction but “how far” is a distance – a scalar – and is positive!
Throwing up is Also Free Fall!
Symmetry of G Field.
v=
v0 + gt
f
Estimate :
a = g ~ 10m / s
2
∆y = v0t +
1 2
gt
2
What Goes Up Must Come Down
Someone standing at the
edge of a cliff throws one
ball straight up and one
straight down at the same
speed. Ignoring air
resistance, which ball
strikes the ground with the
greatest speed?
Free Fall
Question: You throw the rock down with an initial speed of
30 m/s. The rock hits the ground in 3 seconds. With what
speed will the rock hit the ground?
Knowns : v0 =
−30m / s, a =
−9.8m / s 2 , t =
3s
Unknown : v f = ?
v=
v0 + at
f
m
m
=
−30 − 9.8 2 (3s )
s
s
m
v f = −59.4
s
How high is the cliff?
+y
Free Fall
Question: You throw the rock down with an initial speed of
30 m/s. The rock hits the ground in 3 seconds. With what
speed will the rock hit the ground? How high is the cliff?
Knowns : v0 =
−30m / s, a =
−9.8m / s 2 , t =
3s
Unknown : ∆y =
?
1
∆y = v0t + at
+y
2
2
1
=−
( 30m / s )(3s ) + (−9.8m / s )(3s )
= −134m
The cliff is 134 m high.
2
2
2
Free Fall: Throwing Up
What is the speed at the top of the path?
ZERO!
What is the acceleration at the top?
a = -9.80 m/s2
What is the velocity at the same height
on the way down?
-30 m/s
With what velocity will
the rock hit the ground?
-59.4 m/s
SAME as if you threw it
straight down at 30m/s!
+y
Free Fall: Throwing Up Problem
How long does it take to hit the ground? First try to guess!
How long to the top? How long back to launch point? Final v increases by 30m/s?
I guess about 9 seconds!
Knowns : v0 ==
−59.4m / s
30m / s, a −9.8m / s 2 , t =
3s, v f =
Unknown : ∆t =?
v=
v0 + at
f
t
v f − v0 −59.4m / s − 30m / s
=
2
a
−9.8m / s
t = 9.12 s
+y
QuickCheck 2.18
A ball is tossed straight up in the air. At its very
highest point, the ball’s instantaneous acceleration ay
is
A. Positive.
B. Negative.
C. Zero.
Slide 2-96
QuickCheck 2.18
A ball is tossed straight up in the air. At its very
highest point, the ball’s instantaneous acceleration ay
is
A. Positive.
B. Negative.
C. Zero.
Slide 2-97