Projectile Motion

1
Projectile Motion

• Range Formula Proof

2
Range Formula
ax 0
Fx 0
Fy gravity
ay g
Vo
R
Q
3
Initial Velocity Components
Vx ?
V cos Q
Vy
Vy
Vo
Vy ?
V sin Q
Q
Vx
4
Range Formula
V cos Q
V sin Q
V cos Q
-V sin Q
g
0
0
R
R
5
Range Formula
t 2V sin Q
Vfy Voy gt
Solve for time
g
-V sin Q V sin Q gt
sub for time
R Vx t
V cos Q t
( )
R V cos Q
2V sin Q
g
(2 cos Q sin Q) sin 2Q
R V2 sin 2Q
g
6
Range Formula Test Drive
R V2 sin 2Q
g
Unit check?
m
(m/s)2
m
(m2/s2)
(m/s 2)
(m/s 2)
7
Range Formula Test Drive
R V2 sin 2Q
Does it make sense?
g
Test Limits, what ifs
As V goes to 0
R goes to 0
YES!
YES!
As g goes to 0
YES!
YES!
R goes to 0
YES!
Sin 2Q goes to 0
As Q goes to 0
R goes to 0
As Q goes to 90
Sin 2Q goes to 0
R goes to 0
YES!
8
Range Formula Test Drive
R V2 sin 2Q
g
What Q results in the maximum range?
sin 90 1
If Q 45
2Q 90
Rmax at Q 45
9
Range Formula Test Drive
R V2 sin 2Q
g
What 2 angles result in the same Range?
If Q 30
Then 2Q 60
If Q 60
Then 2Q 120
Sin 60 Sin 120
R at 30o R at 60o
10
30 vs 60 Degree Launch