Projectiles
Chapter 5
If we ignore the effects of air resistance an object in the air has a horizontal
acceleration of ax = 0 m/s2 and a vertical acceleration of ay = −9.8 m/s2.
Since ax = 0 the horizontal velocity is constant throughout the flight.
The vertical acceleration of −9.8 m/s2 causes the vertical velocity to decrease on
the way up, be 0 at the peak of the flight and increase in the negative direction
on the way down.
Explanation of the Variables for Projectile Motion
a = −9.8 m/s2
a is vertical acceleration
Vyi Initial Vertical Velocity
Vyf Final Vertical Velocity
yi Initial Height
yf Final Height
t
Time
x Horizontal Distance
𝑉𝑦𝑓 = 𝑉𝑦𝑖 + 𝑎𝑡
If we assume air resistance is
negligible, then Ax horizontal
acceleration is zero.
𝑥 = 𝑉𝑥(𝑡)
(𝑉𝑦𝑓 )2 = (𝑉𝑦𝑖 )2 +2𝑎(𝑦𝑓 − 𝑦𝑖 )
1
𝑦𝑓 = 𝑦𝑖 + 𝑉𝑦𝑖 𝑡 + 𝑎(𝑡)2
2
Vertical Motion Equations
𝑉𝑦𝑓 = 𝑉𝑦𝑖 + 𝑎𝑡
(𝑉𝑦𝑓 )2 = (𝑉𝑦𝑖 )2 +2𝑎(𝑦𝑓 − 𝑦𝑖 )
1
𝑦𝑓 = 𝑦𝑖 + 𝑉𝑦𝑖 𝑡 + 𝑎(𝑡)2
2
Horizontal Motion Equation
𝑥 = 𝑉𝑥(𝑡)
Vertical Motion Equations
𝑉𝑦𝑓 = 𝑉𝑦𝑖 + 𝑎𝑡
• Use to solve for time between
velocity points.
• Use to solve for initial or final
velocity when given time.
Vertical Motion Equations
(𝑉𝑦𝑓 )2 = (𝑉𝑦𝑖 )2 +2𝑎(𝑦𝑓 − 𝑦𝑖 )
Use to find any of the following variables
Vyi Initial Vertical Velocity
Vyf Final Vertical Velocity
yi Initial Height
yf Final Height
Vertical Motion Equations
1
𝑦𝑓 = 𝑦𝑖 + 𝑉𝑦𝑖 𝑡 + 𝑎(𝑡)2
2
1
𝑦𝑓 = 14 + 13 3 + − 9.8(3)2
2
𝑦𝑓 = 14 + 39 − 44.1
𝑦𝑓 = 8.9𝑚
This equation is used to find the height at some time point.
Horizontal Motion Equation
𝑥 = 𝑉𝑥(𝑡)
This equation can be used to find:
• The horizontal distance traveled in some time interval.
• The time to travel a horizontal distance.
• The horizontal velocity needed to travel a distance in a
given time interval.
Takeoff and landing
height effects the
time to the peak
and the time down.
A ball is kicked with an initial vertical velocity (Vyi) of 14 m/s, find
the vertical velocity of the ball after 2.4 seconds.
A football is punted with an initial vertical velocity
(Vyi) of 18 m/s, find the time to the peak.
A golf ball lands with a final vertical velocity (Vyf) of
−23 m/s, find the time down (peak to landing).
A javelin is thrown with an initial height (yi) of 1.3
m, initial vertical velocity (Vyi) of 18 m/s, find the
peak height (yf).
A baseball lands with a final height (yf) of 0 m, final
vertical velocity (Vyf) of −37 m/s, find the peak
height (yi).
A pole vaulter falls from peak height (yi) of 4.1 m to a
final height (yf) of 1.1 m, find the final vertical velocity
(Vyf) of the pole vaulter.
A golf ball is hit with an initial height (yi) of 1.8 m,
initial vertical velocity (Vyi) of 23 m/s, a horizontal
velocity (Vx) of 32 m/s find the height (yf) of the ball
after 3.6 seconds.
Compute the horizontal
distance (x) the ball covers
during the 3.6 seconds.