The horizontal and vertical motions of a projectile are

```Projectile Motion
Objects in projectile motion follow a
parabolic path called a trajectory.
If released at the same instant, a bullet shot from a
gun will hit the ground at the same time as a bullet
dropped from the same height.
The horizontal and vertical motions of a
projectile are independent, meaning they do
not affect each other.
Vertical motion
Horizontal motion
The 2-D motion of a projectile can be separated
into two 1-D motions: horizontal and vertical.
HPM: Horizontal Projectile Motion
The horizontal motion of a projectile is always
constant, if we neglect air resistance.
For projectiles shot at 0°, all of the initial velocity
is in the x direction. Thus, Vyi = 0 m/s.
For projectiles shot at 0°, vertical displacement
and velocity will always be negative.
To hit the target, when should you release
the package?
Rules for Projectile Motion
• Treat horizontal and vertical as two
separate sides of the problems
• TIME is the key, and the only variable that
can be used for both horizontal and
vertical
• Horizontal Motion is always constant
• vx is constant
• ax = 0 m/s2
• Objects follow a parabolic shape
Horizontal Projectile Motion
• All of the initial velocity is in the x direction,
Vyi = 0 m/s
• Vertical displacement and velocity will
always be negative
Example problem
APM: Angled Projectile Motion
For projectiles shot at an angle, initial
velocity is both vertical and horizontal.
Horizontal velocity and initial vertical velocity can
be found using trig functions.
Vi
θ
Vyi
Vx = Vi Cos θ
Vyi = Vi Sin θ
Vx
At the end of the problem, you can recombine
horizontal and vertical velocity to get the total 2-D
velocity.
Vx
Vf
Vyf
Vx
θ
Vf
Vf2 = Vx2 + Vyf2
Vyf
θ = tan-1(Vyf / Vx)
For APM, vy at any height is the same while going
up and coming down except for direction.
Velocities of projectile motion
Note: Vy = 0 at the highest point.
With air resistance, the actual path
is shorter.
Angled Projectile Motion
• Initial velocity is both vertical and
horizontal
• Use trig functions to find vyi and vx
• Vx = Vi Cos θ
• Vyi = Vi Sin θ
• Remember clues
• vy at the top is 0 m/s
• vy at any height is the same while going up
and coming down except for direction
Example Problem
Happy Gilmore hits his shot at 55.0 m/s with an
angle of 50.0° to the ground. How far did the
ball travel before it lands?
•vi = 55.0 m/s
•θ = 50.0°
•ay = -9.81 m/s2
•∆x = ?
vi = 55.0 m/s
θ = 50.0°
• Find vx and vyi
vx
= vi Cos θ
= 55.0 m/s Cos (50.0°)
= 35.4 m/s
Sin θ
= 55.0 m/s Sin (50.0°)
= 42.1 m/s
vyi = vi
ay = -9.81 m/s2
∆x = ?
vi = 55.0 m/s
θ = 50.0°
ay = -9.81 m/s2
vyi = 42.1 m/s
∆x = ?
vx = 35.4 m/s
• What do we need to find ∆x?
• Time! Find time from the vertical side
• ∆y = vyi ∆t + ½ ay ∆t2
0 m = (42.1 m/s) ∆t + ½ (-9.81 m/s2) ∆t2
- (42.1 m/s) ∆t = ½ (-9.81 m/s2) ∆t2
(42.1 m/s) = ½ (9.81 m/s2) ∆t
∆t = 8.58 s
vi = 55.0 m/s
θ = 50.0°
∆t = 8.58 s
ay = -9.81 m/s2
vyi = 42.1 m/s
• Now we can find ∆x
• ∆x = vx ∆t
= (35.4 m/s)(8.58 s)
= 304 m
∆x = ?
vx = 35.4 m/s
Where Should We Aim the Cannon?
At or Above the monkey?
Above the Monkey
At the Monkey
Explanation of monkey experiment
Ranges of projectiles versus angle.
Another interesting application
If speed is great enough…
That’s how the space shuttle and
satellites orbit the earth.
```