Section 3-7: Projectile Motion
A projectile is an
object moving in two
dimensions under the
influence of Earth's
gravity; its path is a
parabola.
Projectile Motion
• Projectile Motion  Motion of an object that is
projected into the air at an angle.
• Near the Earth’s surface, the acceleration a on
the projectile is downward and equal to
a = g = 9.8 m/s2
– Goal: Describe motion after it starts.
• Galileo: Analyzed horizontal & vertical components
of motion separately.
• Today: Displacement D & velocity v are vectors
 Components of motion can be treated separately
Projectile Motion
• Simplest example: Ball rolls across table, to the
edge & falls off edge to floor. Leaves table at time
t = 0. Analyze y part of motion & x part of motion
separately.
• y part of motion: Down is positive & origin is at
table top: y0 = 0. Initially, no y component of
velocity: vy0 = 0

vy = gt, y = (½)g t2
• x part of motion: Origin is at table top: xf = 0. No
x component of acceleration(!): ax = 0. Initially x
component of velocity is: vx

vx = vx0 , x = v0t
Ball Rolls Across Table & Falls Off
t = 0 here

Can be understood by
analyzing horizontal
vertical motions separately.
Take down as positive. Initial
velocity has an x component
ONLY! That is vy0 = 0.
At any point, v has both x &
y components. Kinematic
equations tell us that, at time
t, vx = vx0, vy = gt
x = vx0t
y = (½)gt2
• Summary: Ball rolling across table & falling.
• Vector velocity v has 2 components:
vx = vx0 , vy = gt
• Vector displacement D has 2 components:
x = vx0t , y = (½)g t2
The speed in the x-direction is
constant; in the y-direction the
object moves with constant
acceleration g.
Photo shows two balls that start to
fall at the same time. The one on
the right has an initial speed in the
x-direction. It can be seen that
vertical positions of the two balls
are identical at identical times,
while the horizontal position of the
yellow ball increases linearly.
• PHYSICS: y part of motion:
vy = gt , y = (½)g t2
SAME as free fall motion!!
 An object projected horizontally will reach
the ground at the same time as an object
dropped vertically from the same point!
(x & y motions are independent)
General Case: Object is launched at initial angle θ0 with the
horizontal. Analysis is similar to before, except the initial velocity
has a vertical component vy0  0. Let up be positive now!
vx0 = v0cosθ0 vy0 = v0sinθ0
but, acceleration = g downward for
the entire motion!
Parabolic shape of path is real (neglecting air resistance!)
• General Case: Take y positive upward & origin at
the point where it is shot: x0 = y0= 0
vx = v0cosθ0, vy = v0sinθ0
• Horizontal motion:
NO ACCELERATION IN THE x DIRECTION!
vx = vx0 , x = vx0 t
• Vertical motion:
vy = vy0 - gt ,
y = vy0 t - (½)g t2
(vy) 2 = (vy0)2 - 2gy
– If y is positive downward, the - signs become + signs.
ax = 0, ay = -g = -9.8 m/s2
Summary: Projectile Motion
Projectile motion is motion with constant
acceleration in two dimensions, where the
acceleration is g and is down.
Solving Problems Involving Projectile Motion
1. Read the problem carefully, &choose the object(s) you are
going to analyze.
2. Sketch a diagram.
3. Choose an origin & a coordinate system.
4. Decide on the time interval; this is the same in both directions, &
includes only the time the object is moving with constant
acceleration g.
5. Solve for the x and y motions separately.
6. List known & unknown quantities. Remember that vx never
changes, & that vy = 0 at the highest point.
7. Plan how you will proceed. Use the appropriate equations; you
may have to combine some of them.