1. +

2. +
Motion in Two Dimensions

3. +
Projectile Motion
 Two-dimensional motion is called projectile motion.
 Objects that are thrown or launched into the air and are subject
to gravity are called projectiles.
 Some examples of projectiles are
 Softballs,
 Footballs
 Arrows

4. +
 The parabolic path which is common for all projectile motion is
called trajectory.
 The horizontal distance covered by a projectile which returns its
original height is called the range of the projectile.

5. +
An object is
projected with an
initial velocity, vi, at
an angle of θ.
Resolve the initial
velocity into its x and
ycomponents. Then,
the kinematic
equations can be
applied to describe
the motion of the
projectile throughout
its flight.

6. +
Suppose the initial velocity vector
makes an angle θ with the
horizontal. Again, to analyze the
motion of such a projectile, you
must resolve the initial velocity
vector into its components.
vx,i= vicosθ and vy,i= visinθ

7. +
Example:
(a) Without air resistance, the soccer ball would
travel along a parabola.
(b) With air resistance, the soccer ball would
travel along a shorter path.

8. +
Example:
(a) A long jumper’s velocity while sprinting along the runway can be
represented by a horizontal vector.
(b) Once the jumper is airborne, the jumper’s velocity at any instant
can be described by the components of the velocity.

9. + We can substitute these values for v0x and v0y into the
kinematic equations to obtain a set of equations that
can be used to analyze the motion of a projectile
launched at an angle.
v0x=v0cosθ For the motion on x
x = v0xt = v0cosθt axis
v0y=v0sinθ
vy = v0y – gt
Δy =v0yt – (1/2)gt2 For the motion on y
Δy = (1/2)(v0y+vy)t axis
vy2 = v0y2 – 2gΔy
The relation between
v2 = vx2 + vy2 speeds

10. +
Example
 A ball is launched at a velocity of 8 m/s and an angle of 53o
with the horizontal line axis.
a) Calculate the time for the ball to reach its max. height.
b) How high will the ball rise?
c) What is the range of the ball?
d) What is the ball’s velocity just before it strikes the ground?

11. +
Example
 A rocket is launched at a velocity of 25 m/s and an angle of 37o
with the horizontal line axis.
a) Calculate the time for the ball to reach its max. height.
b) How high will the ball rise?
c) What is the range of the ball?
d) What is the ball’s velocity just before it strikes the ground?