Visualizing Motion In One Dimension
Drawing Motion Diagram
8
1. Draw dots to represent the position of the object at
equal time interval.
2. Draw velocity arrows in the direction of motion.
Relative lengths indicate how fast the object is moving.
v
v2
v1
v
Before leaving the ground, how does the position and
velocity diagrams look like?
3. Draw a velocity change arrow to indicate how the
velocity is changing. Details later.
Think, Answer; Talk to your neighbors, Answer
Two Methods to Draw Velocity Change Arrow
A ball is falling freely, which one of these sketch represents
the situation correctly. 0 represents initial position.
0
A.
C.
B.
D. none of these
Note: the positions are
represented in equal time
interval.

v3
 
v4 v5
Change = Later – Previous ( NOT other way around)
One method ( tip of previous to tip of later):
Step1. Draw two consecutive arrows by lining up their tail.

v2

v1
5
Step2. Draw velocity change from tip of previous to tip of

later.
v2 v

v1
Think, Answer; Talk to your neighbors, Answer
v2 is a later vector compared to v1.
v1
v2
Which one of the following best represents change
in the vector v?
A)
B)
C)
D)
E)

v2

v1
None of the above.
Think, Answer; Talk to your neighbors, Answer
v2 is a later vector compared to v1.
Which one of the following best
represents change in the vector v?
A)
B)
C)
D)
v1
v2
E)
1
x
larger slope -> faster
2
3
4
5
Visualizing Motion With Position vs Time Graph
0
1
x  x0  vxt
when plot is straight.

 x
vx 
t
rise

run
= slope of
x-t plot.
t7
Think, Answer; Talk to your neighbors, Answer
train A
train B
14 km,
20 min
Nine mile
velocity =
displacement
time
B.
C.
D. none of the above.
t
Which graph best depicts the train's velocity vs. time?
v
A
t
v
t
C
v
B
v
D
t
t
Hint: velocity
is the slope of
x – t graph.
Acceleration
Campus
What is average velocity in km/min? direction?
A.
x
Velocity and Speed : Are they same?
26 km,
40 min
t
A train moves along a straight track and its position vs.
time looks like:
Which statement is true:
A. At time tC, both trains have
the same velocity.
time
B. Both trains speed up all the
tC
time.
C. Both trains have the same velocity at some time before
tC .
D. None of the above statements is true.
What is average speed in
km/min?
distance
speed =
time
x
A position vs time graph is
shown here. Which motion
diagram best describes the
nature of the motion?
Think, Answer; Talk to your neighbors, Answer
The graph show positions as a function of time for
two trains running on parallel tracks.
position
Think, Answer; Talk to your neighbors, Answer
Motion diagram of an speeding object

 

v4
v1 v2 v3
v5
v
v5

Let’s make velocity vs time graph
v4



v
change in
v3
a  slope 

velocity.

t
v2
v  v0


v1
t t
or, v  v0  a  t  t0 
or, v  v0  at Assuming t0 = 0.
0
Speed at the
end of time
interval t.
Speed at the
beginning.
t
Direction of acceleration
is same as the direction
of change in velocity
vector.
2
Think, Answer; Talk to your neighbors, Answer
Solve, Answer; Talk to your neighbors, Answer
The velocity vector of a particle moving with constant
acceleration is shown below at two different times, an
earlier time t1 and a later time t2. What is the direction
of the acceleration vector?

v1
A)
B)
E) acceleration is zero
A motor scooter travels east at a speed of 12 m/s. The
driver then reverses direction and heads west at 16 m/s.
It took 4 s to reverse the direction. What is the
acceleration of the scooter?

 v
a
A. 1 m/s2 to the east.
t
B. 1 m/s2 to the west.
C. 7 m/s2 to the east.
D. 7 m/s2 to the west.

v2
C)
E. none of the above.
D)
14
Think, Answer; Talk to your neighbors, Answer
At the moment that the velocity is zero, what is the sign
of the acceleration?
v
a=?
A) Positive
B) negative
t
C) a = zero
Think, Answer; Talk to your neighbors, Answer
v
0
A)
B)
C)
D)
E)
t
0
2
v
An object's velocity
vs. time is:
t
Which graph best represents the object's
acceleration vs. time?
The plot shows velocity vs time graph of a toy car moving
along a horizontal direction. Which choice best describes
motion of the car? Assume positive direction to the right.

Think, Answer; Talk to your neighbors, Answer
A
a
C
t
t
a
B
a
D
t
t
Think, Answer; Talk to your neighbors, Answer
The plot shows velocity vs time graph of a toy car moving
along a horizontal direction. Which choice best describes
motion of the car? Assume positive direction to the right.

+
The car is moving toward the right at constant velocity.
The car is reversing the direction.
The car is moving toward the left at constant velocity.
The acceleration of the car is constant and positive.
The acceleration of the car is increasing.
a
v
0
A)
B)
C)
D)
E)
t
0
2
+
The car is moving toward the right at constant velocity.
The car is reversing the direction.
The car is moving toward the left at constant velocity.
The acceleration of the car is constant and positive.
The acceleration of the car is increasing.
3
Think, Answer; Talk to your neighbors, Answer
The plot shows velocity vs time graph of a toy car moving
along a horizontal direction. Which choice best describes
motion of the car? Assume positive direction to the right.

v
0
A)
B)
C)
D)
E)
t
2
0
Displacement From Velocity vs Time Plot
or, x  vx t
Displacement is product
of velocity and time.
Therefore, displacement is area under the plot.
What is the displacement from 8 to 12 s ?.
vx (m/s)
displacement =
area of rectangle +
area of triangle
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12
t (s)
Displacement From Velocity vs Time Plot: Analytically
vx (m/s)
Displacement = Area under the curve.
x = area of
rectangle + area of
triangle
 (v0 x  0)  t  t0  
vx
v0 x
t0
t
1
(vx  v0 x )  t  t0 
2
t (s)
But, vx  v0 x  axt
1
 v0 x  axt  v0 x  t Assuming t0 = 0.
2 1 2
x  x0  v0 xt  axt Eliminating t from it using vx = v0x+ axt,
2
We get, vx2  v02x  2ax  x  x0  These are called kinematic
equations.
x  v0 xt 

+
The car is moving toward the right at constant velocity.
The car is reversing the direction.
The car is moving toward the left at constant velocity.
The acceleration of the car is constant and positive.
The acceleration of the car is increasing.

 x
vx 
t
Think, Answer; Talk to your neighbors, Answer
The plot shows velocity vs time graph of a toy car moving
along a horizontal direction. Which choice best describes
the motion of the car? positive direction as right.
v
0
A)
B)
C)
D)
E)
t
2
0
+
The car is moving toward the right at constant velocity.
The car is reversing the direction.
The car is moving toward the left at constant velocity.
The acceleration of the car is constant and positive.
The acceleration of the car is increasing.
From Worksheet: How far did the object move in first 8 s?
vx (m/s)
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12
t (s)
What was the average speed in first 8 s?
What was the average acceleration in first 8 s?
Using Kinematic Equations
Problem: Jane and Max are walking towards each other
on a sidewalk. Initially, Jane is 4.0 m east of a lamppost
and Max is 12 m to the east of the same post. If Jane and
Max are walking with a constant speed of 2.0 m/s and 1.2
m/s respectively, when and where would they meet?
Step 1: sketch the situation
Step 2: list known and unknowns with appropriate symbols.
Step 3: Construct equations for both Jane and Max.
Step 4: Solve them and think if the solutions make sense.
4
Think, Answer; Talk to your neighbors, Answer
In a slap shot, a hockey player accelerates the puck from
a velocity of 8.00 m/s to 40.0 m/s in the same direction.
If this shot takes 3.33×10−2s, calculate the distance over
which the puck accelerates.
To solve this problem, which of the following equations
would be most appropriate?
A) vx  v0 x  axt
1
B) x  x0  v0 xt  axt 2
2
C) vx2  v02x  2ax  x  x0 
D) None of these.
E) Can’t be solved in one step. Need two equations.
Freely Falling Objects
0
0.049 m
Positions of freely falling ball every 0.1 s.
Average speed between first two positions,
x  0.049  0  m  0.49m s
vav 

t
 0.1  0  s
It is speed at time (0 + 0.1)/2 = 0.05 s.
t (s)
v (m/s)
vy (m/s)
0.05
0.49
0.15
0.25
1.47
2.45
5
4
3
2
1
0.35
3.43
0.196 m
.2
.3
.4
.5
4.41
t (s)
0.441 m
+y
0.784 m
Can an object have a non-zero acceleration if it has a
zero velocity?
Yes, and I have an example.
Yes, but I can’t think of an example.
No, and I can tell you why.
No, but I’m not sure why.
v0
Suppose rolling uphill
Is velocity increasing or decreasing?
+x
+y
Assume these
coordinates.
What is the direction of change in velocity? sketch first.
Is there acceleration? What is the direction?
Is there acceleration where it is turning Back? What
is direction of the acceleration if any?
Going Up And Coming Down
Highest
y
point

v  ?
slope of red lines
represent velocities.
Down

v  ?
vy
Think, Answer; Talk to your neighbors, Answer
A)
B)
C)
D)
v1
up
0.45
Slope = 9.8 m/s2  g
.1
Motion Along Incline
t
m/s2
a = slope = - 9.8
velocity is zero at
the highest point.
t
Problem: A bullet is fired vertically with initial
speed of 49 m/s. (a) What is its velocity after 7.0 s?
(b) How high is it from the initial position at the 7.0
s? (c) How long it takes for the bullet to arrive at
the initial position?
30
5
Think, Answer; Talk to your neighbors, Answer
A bullet is fired vertically. The bullet goes up and
comes down as shown in the sketch. Which one of
these plot correctly represent the acceleration of
the bullet over time? Assume up as positive
a
A
a
D
a
t
a
t
a
B
t
E
C
t
t
Think, Answer; Talk to your neighbors, Answer
You drop a baseball from the edge of a roof of a building. At
exactly the same time a baseball shoots vertically up from
the surface of the ground towards the woman on the roof in
such a way that the ball just barely reaches the edge of the
roof. Does the ball from the roof reach the ground before the
ball from the ground reaches the roof, or is it the other way
around?
A.
B.
C.
D.
The ball dropped from the roof wins
The ball shot from the ground wins
They tie
More information is needed
Think, Answer; Talk to your neighbors, Answer
Mary drops a stone from a cliff. One second later,
Tom drops and identical stone from the same
location. Ignoring air resistance, what happens to
the distance between the stones as they fall?
A)
B)
C)
D)
It increases
It decreases
It stays the same
It first increases, then decreases
6