In kinematics, the motion of a
particle in a straight line. We distinguished between uniform motion, such as a
billiard ball moving horizontally in a constant velocity; and uniformly
accelerated motion such as a mango falling vertically under the influence of
gravity. Now we are going to consider objects moving in a nonlinear motion or
in a curved path. When you kicked a soccer ball with a certain angle, it
follows a curved path. The path is known as a trajectory. The curved path motion is a combination of a uniform horizontal
motion and a uniformly accelerated vertical motion. Take note that the velocity of the kicked
ball at any instant has two components of motion. The horizontal motion does
not affect the vertical motion. This motion is called projectile motion.
We are fond of playing or watching a
volleyball, football or a basketball match. The ball thrown or kicked in air is an example
of a projectile. A cannonball shot from a cannon, a satellite orbiting the
earth are also projectiles.
To avoid confusion on projectile motion,
let us treat the horizontal component and the vertical component of the motion
separately. If friction is negligible,
the horizontal component of a projectile
motion is similar to a ball
moving a straight line in a constant velocity.
The vertical component of the projectile
motion is similar to a ball moving as a free falling body. The velocity
increases with time.
If we combine these components we will obtain
a projectile motion. A projectile is simply a combination of a uniform
horizontal motion and a uniformly accelerated vertical motion. The only force
acting in the motion is the weight. The path of the projectile motion is known
as the trajectory. When you kicked a ball with a certain angle and it landed to
a certain point, the distance between the kick off point and the landing point
is what you called the range.
For example, you kicked a soccer ball with
an initial velocity of 42 m/s, 37⁰ from the horizontal.
What is the maximum height the ball could reach? How long it will take for the
ball to reach the maximum height? How long is the ball in air? What is the
range?
First you need to get the x and y
component of the velocity, vo,
vox = Vocosθ
vox = 42m/s cos 37⁰
vox = 33.54 m/s
voy = vosinθ
voy = 42 m/s sin 37⁰
voy = 25.28 m/s
To get the height, we are going to use the
equation
h = (v1y 2 - voy2)/
-2g
Remember that at the maximum height the
velocity, v1 is zero. Therefore
h = (v1y 2 - voy2)/
-2g
h = - voy2/ -2g
h = (25.28m/s)2 /-2 (9.8m/s2)
h = 32.6 m/s
To get the time to reach the maximum
height we are going to use the equation
t = (v1y - voy)/ -g
At the maximum height, the velocity v1
is zero. Therefore
t = - voy/ -g
t =-25.28/-9.8 m/s2
t = 2.58s
The time the ball is in the air is
T = 2t
T = 2 (2.58s)
T = 5.16s
To get the range we are going to use the
equation
R = vocosθT
R = (42 cos 30)(5.16s)
R = 173.03 m
Practical Applications:
If you are going to fire a gun
horizontally and drop a bullet vertical with the same height with the gun, at
the same time, which bullet will hit the ground first? Why do we need to avoid
indiscriminate firing?
Predict, Observe, Explain
Put
a coin at a smooth table which is slightly hanged at the edge. Then put another
coin near the overhanging coin. Flick the second coin across the table so that
it strikes the overhanging coin and both coins now will fall on the floor.
Which coin hits the floor first?
