8/29/2011
Summary
Motion: Acceleration
Last time:
• Scalars: Distance, Speed
• Vectors: Position and velocity
Find your clicker!
• Speed = Distance covered/Time taken
• v = Dx / Dt
• Graphs: x vs t and v vs t
Today:
• Graphs: relationship between position and velocity graphs
• Acceleration
Day 3:
• Get your clickers ready!
• Motion
Reminders:
Homework 2 on CULearn
Reading quiz now!
JL in helproom Thursday 9-11am
- Position, velocity
- Acceleration
Relationship between position vs time graph and
velocity vs time graph
• Equations of motion
- Constant velocity
- Constant acceleration
• Changing units
Equations of motion
Several ways to describe motion so far:
1.
2.
3.
4.
Velocity is the slope of position-time graph
+1
Words
Arrows (vectors) and numbers (scalars)
Graphs
Equations
Dx
What does
velocity graph
look like?
Velocity (v) =
0
2
4
time (s)
xf - xi
tf - ti
Rearrange:
xf – xi = v( t f- ti )
xf = xi + v( tf – ti )
-1
Velocity (m/s)
=
Dt
Now let ti = 0s and so xi = x0.
+
xf = xo + v tf
0
2
4
time (s)
The subscript f is now unnecessary so we can write:
x = xo + v t
-
Practice question
Motion at constant velocity
x(t) = x0 + vt
x(t) = x0 + vt
2m/s
0
your position at time t
depends on:
Where you started,
How fast you’re going,
How long you’ve been going
1
Position (m)
I start 1m to the right of the origin at t=0. I walk to the right for 3s at 2m/s.
What is my position at 3s?
(Define positions to the right of the origin as positive)
a.
b.
c.
d.
e.
1m
4m
-5m
3m
7m
1
8/29/2011
There are 1600 meters in a mile. If you
drive 60 miles/hour, how fast is this in m/s?
Break to discuss units
If you drive 60 miles/hour, that’s a speed.
It’s also 1 mile/minute
It’s also 1/60 mile/s
a)
b)
c)
d)
e)
“Physics” units: meters/second (m/s)
60 m/s
160 m/s
27 m/s
270 m/s
1600 m/s
How did you get that?
How did you get that?
• We want to change the units but keep the answer (speed) the same
• We want to change the units but keep the answer (speed) the same
• Remember 2 things:
- Multiply any answer by 1 and it doesn’t change
- 1600m = 1mile  1600m
=1
1 mile
• Remember 2 things:
- Multiply any answer by 1 and it doesn’t change
- 1600m = 1mile  1600m
=1
1 mile
• We have speed in mi/hr, we want m/s so:
First change miles to meters
 60 1600m 

  27m / s
 60  60s 
Position
Then change hours to seconds
+
0
time
Velocity
Demo 4:
Sketch position vs time and
velocity vs time graphs for
when Moving Man:
walks steadily towards the
tree for 6 seconds,
then stands still for 6
seconds, and
then towards the house twice
as fast as before for 6
seconds.
 60mi   1600m    1hr   1 min 
 


 
 
 hr   1mi   60 min   60s 
+
0
time
Walking man moves according to the
distance-time graph, below. At which time is
Walking Man slowing down?
a) A only
b) B only
c) C only
+
d) A and C
e) A, B, and C
position
Speed =
Speed =
 60mi 


 hr 
0
A
B
C
time
-
(start at v = -0.8 m/s)
5s
10 s
15 s
20 s
2
8/29/2011
Equations when velocity is changing
Graph shows the velocity of a car as a function of time.
What is its acceleration?
a. -0.25m/s2
b. 0.25m/s2
c. 0.4m/s
d. 0.25m/s
e. -0.65m/s2
What if velocity is changing? … Accelerating
Acceleration (a) = Gradient of a velocity vs time graph
=
Change in velocity (Dv)
Time taken (Dt)
vf - vi
=
tf - ti
Velocity (m/s)
Units = m/s = m/s2
s
+5
0
time
-8
5s
Equations when velocity is changing
15 s
20 s
Motion at constant acceleration
What if velocity is changing? … Accelerating
Acceleration (a) =
10 s
vf - vi
tf - ti
v(t) = v0 + at
Rearrange:
vf – vi = a( t f- ti )
vf = vi + a( tf – ti )
your velocity at time t
depends on
Now let ti = 0s and so vi = v0 and drop the f subscript
v = v0 + a t
Your velocity when
you started,
How fast you’re accelerating,
How long you’ve been going
What about position at constant acceleration?
So far: x = x0 + vt (a = 0)
From (1) :
v = v0 + at (a = constant)
(2)
(if a ≠ 0)
(3)
x = x0 + vaveraget
Position at constant acceleration
(1)
x(t) = x0 + v0 t + ½ at2
vaverage = v0+ ½ (change in velocity)
a = change in velocity / time taken
vaverage= v0 + ½ at
Substitute (4) into (3)
x = x0+ (v0 + ½ at)t
x = x0 + v0t + ½ at2
(4)
your position at time t
depends on
Where you started,
and
How fast you started going,
and
How fast you’re accelerating,
and
How long you have
been going
3
8/29/2011
position
A car accelerates at a steady rate from stationary along a straight road.
Sketch position, velocity and acceleration as a function of time
Hint:
a is gradient of v vs t
v is gradient of x vs t
+
0
time
Velocity
+
0
time
Acceleration
+
0
time
-
4