Flight of Projectile
A projectile is an object that is given some energy to cause it to go up, reach a maximum
height, and then fall to the ground. The projectile can be a soccer ball, rock, bullet,
baseball, or cannon ball. These objects all take on the same characteristic arc as they fly
through the air. Soccer balls get kicked, rocks and baseballs get tossed, and bullets and
cannon balls all fly through the air with a characteristic upside-down u-shape.
The flight of a projectile depends on three variables in order to model its flight through
space: the gravity on Earth, the starting height of the projectile, and the amount of
vertical force given to launch the projectile.
Gravity
Earth's gravity is different than the gravity on other planets. On
Earth, gravity pulls on objects, fighting their ability to reach high
heights and bringing them back down to the ground. Since it is a
downward force, we think of it as a negative force. That force is
32 feet/sec every second.
Velocity
The starting upward velocity will differ for each problem. The
more upward velocity we give it, the better it can fight against
gravity, and the higher the projectile will go before it falls back
down.
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Initial Height
(Distance)
The initial height of the projectile is important. If the projectile
starts up high, it can get even higher after it is launched.
y = height of the object in a given amount of time
= x(tanØ) -
𝑔𝑥 2
2𝑉 2 (cos ∅)2
1
or V(sinØ)(t) - (g)(𝑡)2
2
x = horizontal distance of the object in a given amount of time
= V ( cosØ ) (t)
H = maximum height of the object
=
𝑽𝟐 (𝐬𝐢𝐧 ∅)𝟐
𝟐𝒈
R = maximum horizontal distance
=
𝑽𝟐 𝐬𝐢𝐧 𝟐∅
𝒈
VB = velocity of the object in a given amount of time
= √𝑽𝑩𝒙 𝟐 + 𝑽𝑩𝒚 𝟐
Tangential and Normal components of Acceleration
The strong accelerations experienced in a roller coaster are not only due to the changes
of the speed, but also to the curved trajectory. The rate of change of the speed is only
one of the components of the acceleration, namely, the tangential component. The other
component of the acceleration depends on the curvature of the trajectory as it is shown
in this chapter.
at = tangential acceleration
an = normal acceleration =
𝑉2
𝑟
= rω2
V = velocity at any point which is tangent to the path = rω
ω = angular speed in rad/sec
a = Total acceleration = √(𝑎𝑡 2 ) + (𝑎𝑛 2 )
r = radius of curvature at any point
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an = ay cos Ɵ – ax cos Ɵ
Centrifugal Force
Centrifugal force is an outward force apparent in a rotating reference frame. It does not
exist when a system is described relative to an inertial frame of reference.
In a reference frame rotating about an axis through its origin, all objects, regardless of
their state of motion, appear to be under the influence of a radially (from the axis of
rotation) outward force that is proportional to their mass, to the distance from the axis
of rotation of the frame, and to the square of the angular velocity of the frame. This is the
centrifugal force
Centrifugal force =
𝑊 𝑉2
𝑔𝑟
or
𝑚𝑣 2
𝑟