Falling Objects and Projectile
Motion
Physics 1010
13 June 2012

Acceleration Due to Gravity
Falling objects accelerate to the earth due to gravitational attraction.
The Relationship is the same as for any acceleration.
The acceleration of gravity is a vector pointing downward.
 v d
 a



 v o
 v o g


 a
1
2 t
 a t
2
The sign depends on reference: up or down.
g g



9 .
8
9 .
8 m s
2 m s
2
 up
 down

Measuring Gravitational Acceleration
Galileo’s method: Use your pulse as a timer and an inclined plane to slow the acceleration in proportion to the angle of slope.
Stroboscopic light: The action in the photo stops at predetermined intervals (Δt = 0.05 sec in book illustration). Measure distances from the photo and scale to actual distances.
Timer and track: Use a timer instead of pulse on specific distances down the track. Our lab did this.

Aristotle and Galileo
How were their results different? Why?

Dropping a Ball
Using the equations: v
 d


 v o
 v o t


 a
1 t
2
 a t
2 v
 up
 d
 up

 v o v o
 up
 up

 t
(

 g
 up
1
2

 g
) t

 up
( v o
 t
2
 gt )
 up d
 v o t

1
2 gt
2
We can solve all problems with a falling object.

v o t 1
2 gt

Tossing a ball upward
Using the equations: v
 v o

 a t v
 up
 v o
 up

(
 g
 up ) t v d

 v o
 v o d
 up t
 gt



1 v o
2
 a t
2
 up
 t

1
2


 g
 up
( v o
 t
2
 gt )
 up d
 v o t

1
2 gt
2
We can do any problem with a ball tossed upward.
We just need to watch the directions.

Facts
Initial height up
Initial velocity g up up
Values
0
10
-9.8
0.2
10.0
5.0
0.0
-5.0
0.0
-10.0
-15.0
-20.0
Final Height
1.0
2.0
Final Height
3.0
Ball Toss Up
4.0
Initial time(sec) Height
0.0
0.0
0.2
0.4
0.6
0.8
0.0
0.0
0.0
0.0
1.0
1.2
1.4
0.0
0.0
0.0
2.4
2.6
2.8
3.0
3.2
1.6
1.8
2.0
2.2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
v o t
16.0
18.0
20.0
22.0
24.0
26.0
28.0
30.0
32.0
8.0
10.0
12.0
14.0
0.0
2.0
4.0
6.0
1
2 gt equals
0.0 equals
-0.2 equals
-0.8 equals
-1.8 equals
-3.1 equals
-4.9 equals
-7.1 equals
-9.6 equals
-12.5 equals
-15.9 equals
-19.6 equals
-23.7 equals
-28.2 equals
-33.1 equals
-38.4 equals
-44.1 equals
-50.2 equals
Final
Height
0.0
1.8
3.2
4.2
4.9
5.1
4.9
4.4
3.5
2.1
0.4
-1.7
-4.2
-7.1
-10.4
-14.1
-18.2

Facts
Initial height up
Initial velocity g up up
Values
0
20
-9.8
0.2
25.0
20.0
15.0
10.0
5.0
0.0
0.0
Final Height
1.0
2.0
Final Height
3.0
Ball Toss Up
4.0
Initial time(sec)
0.0
0.2
Height
0.0
0.0
0.4
0.6
0.8
1.0
0.0
0.0
0.0
0.0
2.0
2.2
2.4
2.6
1.2
1.4
1.6
1.8
2.8
3.0
3.2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
v o t
32.0
36.0
40.0
44.0
48.0
52.0
56.0
60.0
64.0
0.0
4.0
8.0
12.0
16.0
20.0
24.0
28.0
1
2 gt equals
0.0 equals
-0.2 equals
-0.8 equals
-1.8 equals
-3.1 equals
-4.9 equals
-7.1 equals
-9.6 equals
-12.5 equals
-15.9 equals
-19.6 equals
-23.7 equals
-28.2 equals
-33.1 equals
-38.4 equals
-44.1 equals
-50.2 equals
Final
Height
0.0
3.8
7.2
10.2
12.9
15.1
16.9
18.4
19.5
20.1
20.4
20.3
19.8
18.9
17.6
15.9
13.8

Projectile Motion
Using these equations for the up (y) direction: v
 y

 v o y

 a y t ; g

9 .
8 m s
2 v
 y
 v oy
 y

(
 g
 y ) t

( v oy
 gt )
 y v
 v oy d
 y

 gt
 v o
 y
 t

1
2

 g
 y
 t
2 d y
 v o y t

1
2 gt
2
Using these equations for the horizontal (x) direction v
 x
 v ox
; d x
 v ox t
We have it all!

Facts
Initial height up
Initial velocity g up up
Values
0
20
-9.8
0.2
Velocity
Right (x)
10 Cannon Ball Flight x
 v o t
Initial time(sec) X motion
0.0
0.2
0.0
2.0
Height
0.0
0.0
4.2
4.4
4.6
4.8
5.0
5.2
5.4
5.6
5.8
6.0
3.2
3.4
3.6
3.8
4.0
2.0
2.2
2.4
2.6
2.8
3.0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
42.0
44.0
46.0
48.0
50.0
52.0
32.0
34.0
36.0
38.0
40.0
54.0
56.0
58.0
60.0
20.0
22.0
24.0
26.0
28.0
30.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
v o t
20.0
24.0
28.0
32.0
36.0
0.0
4.0
8.0
12.0
16.0
1
2 gt equals
0.0 equals
-0.2 equals
64.0
68.0
72.0
76.0
80.0
40.0
44.0
48.0
52.0
56.0
60.0
-50.2 equals
-56.6 equals
-63.5 equals
-70.8 equals
-78.4 equals
84.0
88.0
92.0
-86.4 equals
-94.9 equals
-103.7 equals
96.0
-112.9 equals
100.0
-122.5 equals
104.0
-132.5 equals
108.0
-142.9 equals
112.0
-153.7 equals
116.0
-164.8 equals
120.0
-176.4 equals
-0.8 equals
-1.8 equals
-3.1 equals
-4.9 equals
-7.1 equals
-9.6 equals
-12.5 equals
-15.9 equals
-19.6 equals
-23.7 equals
-28.2 equals
-33.1 equals
-38.4 equals
-44.1 equals
13.8
11.4
8.5
5.2
1.6
20.4
20.3
19.8
18.9
17.6
15.9
Final
Height
0.0
3.8
7.2
10.2
12.9
15.1
16.9
18.4
19.5
20.1
-2.4
-6.9
-11.7
-16.9
-22.5
-28.5
-34.9
-41.7
-48.8
-56.4

Max Distance
Finding initial components of velocity: v x
 v cos
 v v y
 v sin
 v y
Θ v x