Projectile Motion
• A projectile is anything launched, shot or
thrown---i.e. not self-propelled. Examples:
a golf ball as it flies through the air, a
kicked soccer ball, a thrown football, a
speeding bullet.
• The path that the projectile follows is
called a trajectory.
2-D Projectile Motion: Basic Equations
Assumptions:
• ignore air resistance
• g = 9.80 m/s2, downward
• ignore Earth’s rotation
If y-axis points upward, acceleration in
x-direction is zero and acceleration in
y-direction is -9.80 m/s2
2-D Projectile Motion
𝑥𝑓 = 𝑥𝑖 + 𝑣𝑥𝑖𝑡
𝑦𝑓 = 𝑦𝑖 + 𝑣𝑦𝑖𝑡 −1
2𝑔𝑡2
𝑣𝑥𝑓 = 𝑣𝑥𝑖
𝑣𝑦𝑓 = 𝑣𝑦𝑖 − 𝑔𝑡
Where g = 9.80 m/s2.
This motion is for a passive projectile. A
missile or vehicle (e.g. airplane) may also
have self-controlled acceleration (ax, ay, az).
2-D Projectile Motion
𝑥 = 𝑥𝑖 + 𝑣𝑥𝑡
𝑦 = 𝑦𝑖 + 𝑣𝑦𝑖𝑡 −1
2𝑔𝑡2
Motion in the x-y plane will be parabolic,
because horizontal position is linear with
time, and vertical position is quadratic.
𝑡 = 𝑥 − 𝑥𝑖 𝑣𝑥
𝑦 = 𝑦𝑖 + 𝑣𝑦𝑖 𝑥 − 𝑥𝑖 𝑣𝑥 −1
2𝑔 𝑥 − 𝑥𝑖 𝑣𝑥
2
𝑦 = 𝐶 + 𝐵𝑥 + 𝐴𝑥2
A dropped object and
an object launched
horizontally will fall at
the same rate. Motion
in the horizontal and
vertical directions are
independent.
The vertical motion
dictates the “time of
flight”.
2-D Projectile Motion
Demonstration!
A new extreme sport?
(a) Write the horizontal and vertical equations of
motion for v0 = 1.30 m/s. (b) What is the x and y
position of the basketball at t = 0.500 s?
Two friends jump at the same time.
(a) Who hits the water first? (b) Who
hits the water with greater speed?
Launch Angle
Launch angle: direction of initial velocity with
respect to horizontal. This affects the
location of the vertex.
2-D Projectile Motion
If an object is launched at an initial angle of θ0
with the horizontal, the analysis is similar except
that the initial velocity has a vertical component.
Launch Angle
For angle θ, v0x = v0 cos θ and v0z = v0 sin θ.
This gives the equations of motion:
𝑥𝑓 = 𝑥𝑖 + 𝑣0 cos 𝜃 𝑡
𝑦𝑓 = 𝑦𝑖 + 𝑣0 sin 𝜃 𝑡 −1
2𝑔𝑡2
𝑣𝑥 = 𝑣0 cos 𝜃 = constant
𝑣𝑦𝑓 = 𝑣0 sin 𝜃 − 𝑔𝑡
Range
The range of a projectile is the horizontal
distance it travels before landing:
Range = 𝑥𝑓 − 𝑥𝑖 = 𝑣0 cos 𝜃 𝑡
𝑦𝑓 = 𝑦𝑖 + 𝑣0 sin 𝜃 𝑡 −1
2𝑔𝑡2
Solve the y- equation for t, defining ∆𝑦 = 𝑦𝑓 − 𝑦𝑖
𝑡 =𝑣0 sin 𝜃 ± 𝑣0 sin 𝜃
2 − 2𝑔∆𝑦
𝑔
Range = 𝑣0 cos 𝜃𝑣0 sin 𝜃± 𝑣0 sin 𝜃 2−2𝑔∆𝑦
𝑔
Let’s consider special cases 𝜃 = 0 and ∆𝑦 = 0 .
Range: Special Case ∆𝑦 = 0
Range = 𝑣0 cos 𝜃𝑣0 sin 𝜃+ 𝑣0 sin 𝜃 2+0
𝑔=
𝑣02 2 cos 𝜃 sin 𝜃
𝑔=𝑣0
2 sin 2𝜃
𝑔
Maximum range occurs
when
sin 2𝜃 = 1i.e.
𝜃 = 45°then
𝑅 = 𝑣02 𝑔
Reality Check
Air resistance causes horizontal and vertical
velocities to decelerate.
Wind also affects the range of a projectile.
Spin on a projectile also affects its motion
(“lift” due to relative velocity + Bernoulli’s
Principle)
Projectiles get their initial
velocity from…
• Elasticity – thrown objects, bow, slingshot,
torsion catapult, toy spring dart gun
• Rotational motion - sling
• Gravity – trebuchet, ramp (e.g. ski jump)
• Expanding gas – blow gun, cannon, rifle
• Electromagnetic force – rail gun, coil gun
Self-powered rockets and guided missiles
are not projectiles.
Hitting a Target
If the rifle is fired directly at the target in a horizontal direction, will the bullet hit the center of the target?
Does the bullet fall during its flight?
Which of the two trajectories shown will result in a longer time for the ball to reach home plate?
a) The higher trajectory.
b) The lower trajectory.
c) They will take the same time.
a) The higher trajectory takes
longer. The time of flight is
determined by the initial vertical
velocity component which also
determines the maximum height
reached.
ConcepTest 4.2 Dropping a Package
You drop a package from
a plane flying at constant
speed in a straight line.
Without air resistance, the
package will:
1) quickly lag behind the plane
while falling
2) remain vertically under the
plane while falling
3) move ahead of the plane while
falling
4) not fall at all
You drop a package from
a plane flying at constant
speed in a straight line.
Without air resistance, the
package will:
1) quickly lag behind the plane
while falling
2) remain vertically under the
plane while falling
3) move ahead of the plane while
falling
4) not fall at all
Both the plane and the package have
the same horizontal velocity at the
moment of release. They will maintain
this velocity in the x-direction, so they
stay aligned.
Follow-up: What would happen if air resistance were present?
ConcepTest 4.2 Dropping a Package