find time of flight , maximum height , horizontal component of projectile
Answers
Basic Equations and Parabolic Path
Projectile motion is a form of motion where an object moves in parabolic path; the path that the object follows is called its trajectory.
LEARNING OBJECTIVES
Assess the effect of angle and velocity on the trajectory of the projectile; derive maximum height using displacement
KEY TAKEAWAYS
Key Points
Objects that are projected from, and land on the same horizontal surface will have a vertically symmetrical path.
The time it takes from an object to be projected and land is called the time of flight. This depends on the initial velocity of the projectile and the angle of projection.
When the projectile reaches a vertical velocity of zero, this is the maximum height of the projectile and then gravity will take over and accelerate the object downward.
The horizontal displacement of the projectile is called the range of the projectile, and depends on the initial velocity of the object.
Key Terms
trajectory: The path of a body as it travels through space.
symmetrical: Exhibiting symmetry; having harmonious or proportionate arrangement of parts; having corresponding parts or relations.
Projectile Motion
Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In a previous atom we discussed what the various components of an object in projectile motion are. In this atom we will discuss the basic equations that go along with them in the special case in which the projectile initial positions are null (i.e.
x
0
=
0
and
y
0
=
0
).
Initial Velocity
The initial velocity can be expressed as x components and y components:
u
x
=
u
⋅
cos
θ
u
y
=
u
⋅
sin
θ
In this equation,
u
stands for initial velocity magnitude and
θ
refers to projectile angle.
Time of Flight
The time of flight of a projectile motion is the time from when the object is projected to the time it reaches the surface. As we discussed previously,
T
depends on the initial velocity magnitude and the angle of the projectile:
T
=
2
⋅
u
y
g
T
=
2
⋅
u
⋅
sin
θ
g
Acceleration
In projectile motion, there is no acceleration in the horizontal direction. The acceleration,
a
, in the vertical direction is just due to gravity, also known as free fall:
a
x
=
0
a
y
=
−
g
Velocity
The horizontal velocity remains constant, but the vertical velocity varies linearly, because the acceleration is constant. At any time,
t
, the velocity is:
u
x
=
u
⋅
cos
θ
u
y
=
u
⋅
sin
θ
−
g
⋅
t
You can also use the Pythagorean Theorem to find velocity:
u
=
√
u
2
x
+
u
2
y
Displacement
At time, t, the displacement components are:
x
=
u
⋅
t
⋅
cos
θ
y
=
u
⋅
t
⋅
sin
θ
−
1
2
gt
2
The equation for the magnitude of the displacement is
Δ
r
=
√
x
2
+
y
2
.
Parabolic Trajectory
We can use the displacement equations in the x and y direction to obtain an equation for the parabolic form of a projectile motion:
y
=
tan
θ
⋅
x
−
g
2
⋅
u
2
⋅
cos
2
θ
⋅
x
2
Maximum Height
The maximum height is reached when
v
y
=
0
. Using this we can rearrange the velocity equation to find the time it will take for the object to reach maximum height
t
h
=
u
⋅
sin
θ
g
where
t
h
stands for the time it takes to reach maximum height. From the displacement equation we can find the maximum height
h
=
u
2
⋅
sin
2
θ
2
⋅
g
Range
The range of the motion is fixed by the condition
y
=
0
. Using this we can rearrange the parabolic motion equation to find the range of the motion:
R
=
u
2
⋅
sin
2
θ
g
.
image
Range of Trajectory: The range of a trajectory is shown in this figure.
Time of Flight Formula
A projectile is an object that is given an initial velocity, and is acted on by gravity. The amount of time it spends in the air is called the time of flight. If the ground from which the projectile is launched is level, the time of flight only depends on the initial velocity v0, the launch angle θ, and the acceleration due to gravity. The unit for the time of flight is seconds .
t=2v sinθ÷g
t = time of flight (s)
v = initial velocity (m/s)
g = acceleration due to gravity (9.80 m/s2)
θ = angle of the initial velocity from the horizontal plane (radians or degrees)