Galileo Galilei was the first to make an analysis
of the motion of freely falling objects.
Based on his experimentations and reasoned
deductions, Galileo concluded that all objects
on earth, in the absence of air resistance
would fall with the same constant
acceleration due to gravity.
t = /2d/g
a = (vf – vi)/t
d = [(vi + vf)t]/2
where g is the acceleration due to gravity
(980cm/sec2 or 9.8 m/sec2)
A stone is dropped into the river from a bridge 7.0
high. How long would it take before the stone
crashes into the water?
Given: d = 7.0m
g = 9 .8 m/s2
t = /2d/g
/2(7.0 m)/9.8 m/s2
t = 1.2 s
What is the bullet’s speed at the peak of its path?
Given: vi = 300 m/s
a = -9.8m/s2
The speed of the bullet at the peak of its path is
zero since that is the point the object is at rest and
has lost all speed it initially has due to deceleration.
What is its deceleration on its way up? At the peak?
Its acceleration on its way down?
The deceleration of the object on its way up is
9.8m/s2. At the peak it is 9.8 m/s2 downward. And
on its way down the object increases its speed and
therefore accelerates with a rate of 9.8 m/s2. It is
important to note the acceleration due to gravity is
taking effect at any point during the object’s flight.
How high is this peak?
To solve for the height of the peak we
concentrate on the upward motion until the
object reaches the peak. During this
segment the initial speed vi = 300 m/s.
The final speed vf = 0 since the object is at
rest at the peak. The acceleration is -9.8
m/s2. We use the equation d = [(vi + vf)t]2
to solve for the height where t can be solved
using a = (vf – vi)/t.
t = (vf – vi)/a = (300m/s - 0)/-9.8m/s2
= 31 sec
d = [(vi + vf)t]/2 = [(300m/s + 0)31s]/2
= 4650 meters
Find the time the bullet will take to reach its
starting point from the moment it is shot?
The total time of flight which is the time to
go up plus the time to go down is twice 31s
(the value going up) equal to 62 seconds.
A ball is dropped from the window of a
three-storey apartment. How far has it
fallen after one second? What is the ball’s
velocity after one second of fall? How far
does the ball fall during the next second?
Assume no air resistance.
Carlo threw a ball upward with an initial velocity
of 2000 cm/s and was able to catch it before it
reached the ground on its return.
a. What was its velocity after 1s?
b. What was its displacement in the first
c. How long did it take the ball to reach its
d. How far was the maximum height from the
e. What was its final velocity just before it
touched Carlo’s hand?
f. How long will it take the ball to reach a point
1000 cm above Carlo’s hand on its way down?
A car was moving at 70 m/h when the driver
saw a boy crossing the road. The driver was
able to stop the car in 3.8sec and just
avoided hitting the boy. Determine:
a. thinking distance if the driver’s reaction
time is 0.75 sec.
b. uniform deceleration when brakes were
c. stopping distance of the car
A ball is thrown straight up so that it reaches a
maximum height of 30m.
1. How long did it take the ball to go up?
2. How fast was it going when it was 20m high
from the ground?
Projectile – an object or body thrown in a
curved path given an initial velocity and
whose motion is influenced by the pull of
Trajectory – the path of a projectile that is
principally determined by two types of
motion (horizontal and vertical motion)
Projectile motion – motion of a body thrown
in a curved path with constant acceleration
Distance is total
(range) d = dx
Motion is accelerated
due to gravity
Distance is height
t = √(2d)/g
t = dx/v
Suppose that a coin rolls off the edge of a table
1.5 m high at a horizontal speed of 1.0 m/s.
a. Find the time it would take before it lands
on the floor
b. How far from the base of the table will it
2. A little girl throws her jackstone ball
horizontally out of the window with a velocity
of 30m/sec. If the window s 3.0 meters above
the ground level, how far will the ball travel
before it hits the ground?
R = [vo2 sin θ]/g = vo2 [2sin θ cos θ] /g
T = [2vo sin θ]/g
H = [vo2 sin2θ]/2g
1. An arrow leaves a bow at 30 m/sec at an
angle 30o from the horizontal.
a. What is its maximum range?
b. Find the maximum height of the arrow.
c. How long would the arrow take to land on
the other side exactly opposite where it is
2. A projectile is thrown with a speed of 100
m/sec in a direction 30o above the
horizontal. Find the height to which it rises,
the time of flight, and the horizontal range.