Top-Rated Free Essay
Preview

Orbit and Maximum Height

Good Essays
739 Words
Grammar
Grammar
Plagiarism
Plagiarism
Writing
Writing
Score
Score
Orbit and Maximum Height
Projectile motion is a form of motion where a particle (called a projectile) is thrown obliquely near the earth's surface, & it moves along a curved path under the action of gravity. The path followed by a projectile motion is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning of the trajectory after which there is no interference apart from gravity.
The initial velocity

If the projectile is launched with an initial velocity v0, then it can be written as

\mathbf{v}_0 = v_{0x}\mathbf{i} + v_{0y}\mathbf{j}.

The components v0x and v0y can be found if the angle, ϴ is known:

v_{0x} = vgh_0\cos\theta, v_{0y} = v_0\sin\theta.

If the projectile's range, launch angle, and drop height are known, launch velocity can be found by

V_0 = \sqrt{{R^2 g} \over {R \sin 2\theta + 2h \cos^2\theta}} .

The launch angle is usually expressed by the symbol theta, but often the symbol alpha is used.
Kinematic quantities of projectile motion

In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other.
Acceleration

Since there is no acceleration in the horizontal direction velocity in horizontal direction is constant which is equal to ucosα. The vertical motion of the projectile is the motion of a particle during its free fall. Here the acceleration is constant, equal to g.[1] The components of the acceleration:

a_x = 0 , a_y = -g .

Velocity

The horizontal component of the velocity remains unchanged throughout the motion. The vertical component of the velocity increases linearly, because the acceleration is constant. At any time t, the components of the velocity:

v_x=v_0 \cos(\theta) , v_y=v_0 \sin(\theta) - gt .

The magnitude of the velocity (under the Pythagorean theorem):

v=\sqrt{v_x^2 + v_y^2 \ } .

Displacement
Displacement and coordinates of parabolic throwing

At any time t, the projectile's horizontal and vertical displacement:

x = v_0 t \cos(\theta) , y = v_0 t \sin(\theta) - \frac{1}{2}gt^2 .

The magnitude of the displacement:

\Delta r=\sqrt{x^2 + y^2 \ } .

Parabolic trajectory

Consider the equations,

x = v_0 t \cos(\theta) , y = v_0 t \sin(\theta) - \frac{1}{2}gt^2 .

If we eliminate t between these two equations we will obtain the following:

y=\tan(\theta) \cdot x-\frac{g}{2v^2_{0}\cos^2 \theta} \cdot x^2,

This equation is the equation of a parabola. Since g, α, and v0 are constants, the above equation is of the form

y=ax+bx^2,

in which a and b are constants. This is the equation of a parabola, so the path is parabolic. The axis of the parabola is vertical.
The maximum height of projectile
Maximum height of projectile

The highest height which the object will reach is known as the peak of the object's motion. The increase of the height will last, until v_y=0, that is,

0=v_0 \sin(\theta) - gt_h .

Time to reach the maximum height:

t_h = {v_0 \sin(\theta) \over g} .

From the vertical displacement the maximum height of projectile:

h = v_0 t_h \sin(\theta) - \frac{1}{2}gt^2_h h = {v_0^2 \sin^2(\theta) \over {2g}} .

Additional Equation

For the relation between the distance traveled, the maximum height and angle of launch, the equation below has been developed.

h= {d \tan(\theta) \over {4}}
References

Budó Ágoston:, összefüggések és adatok, Budapest, Nemzeti Tankönyvkiadó, 2004. ISBN 963-19-3506-X (Hungarian)

Notes

^ The g is the acceleration due to gravity. (9.81 m/s2 near the surface of the Earth).

External links

Open Source Physics computer model
A Java simulation of projectile motion, including first-order air resistance

NOTE: Since the value of g is not specific the body with high velocity over g limit cannot be measured using the concept of the projectile motion.

A projectile is any object upon which the only force is gravity,
Projectiles travel with a parabolic trajectory due to the influence of gravity,
There are no horizontal forces acting upon projectiles and thus no horizontal acceleration,
The horizontal velocity of a projectile is constant (a never changing in value),
There is a vertical acceleration caused by gravity; its value is 9.8 m/s/s, down,
The vertical velocity of a projectile changes by 9.8 m/s each second,
The horizontal motion of a projectile is independent of its vertical motion.

You May Also Find These Documents Helpful

  • Good Essays

    This experiment was to use kinetics of projectile motion and free falling bodies to determine the distance a ball will travel after it hits a bounce plate. To determine this we had to use the equations x=(1/2)at2 and v=v0+at and derive an equation that will determine the distance the ball will travel based on the height of the bounce plate and the height of where the ball will be dropped above the bounce plate. The equation made was g*(sqrt(2)/sqrt(g))*(sqrt(H)*sqrt(h)). From here we can make an estimate of how far the ball will travel after it hits the bounce plate.…

    • 1086 Words
    • 5 Pages
    Good Essays
  • Good Essays

    Before working on this activity, you should view the movie entitled <Galileo's Projectile_15fps.mov> and review the definitions of two-dimensional displacement and velocity vectors.…

    • 1266 Words
    • 4 Pages
    Good Essays
  • Good Essays

    Acceleration and Ball

    • 277 Words
    • 2 Pages

    A ball is thrown upward at time t = 0 s from the ground with an initial velocity of 4 m/s (~ 9 mph). Assume that g = 10 m/s2.…

    • 277 Words
    • 2 Pages
    Good Essays
  • Good Essays

    Physics 101 Study Guide

    • 5360 Words
    • 22 Pages

    EXAMPLE 5: A ball is thrown vertically upward from a window that is 3.6 m above the ground. The ball's initial speed is 2.8 m/s and the acceleration due to gravity is 9.8 m/s2.…

    • 5360 Words
    • 22 Pages
    Good Essays
  • Satisfactory Essays

    Psy094 Unit 6 Lab Report

    • 503 Words
    • 3 Pages

    as shown in Figure 2. Determine the vertical drop, Sy of the bullet after travelling…

    • 503 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    16. How fast must a projectile moving horizontally travel so that the curve it follows matches the curve of Earth? If it is close to Earth then it is around 8km/s.…

    • 349 Words
    • 2 Pages
    Satisfactory Essays
  • Good Essays

    What?

    • 3646 Words
    • 15 Pages

    What dimension controls time in falling body and projectile motion problems? - The vertical dimension.…

    • 3646 Words
    • 15 Pages
    Good Essays
  • Good Essays

    Pros And Cons Of Rifling

    • 1021 Words
    • 5 Pages

    Bullets should travel in a parabolic path; however, there are a number of factors which affect the path of a bullet. These include the forces acting on the bullet, twist rate and gyroscopic rate.…

    • 1021 Words
    • 5 Pages
    Good Essays
  • Good Essays

    Projectile and Marble

    • 1220 Words
    • 7 Pages

    You have learned that the motion of any object moving through the air affected only by gravity is an example of projectile motion. Examples of projectile motion include a basketball thrown toward a hoop, a car driven off a cliff by a stunt person, and a marble launched from the CPO marble launcher. Toss a ball some distance and you can imagine in your mind the arc it follows, first rising then falling to the ground. Projectile motion is also called two-dimensional motion because it depends on two components: vertical and horizontal. The range is the horizontal distance that the projectile travels between launch and landing. In this experiment, you will see how the range is related to the launch angle and initial velocity of launch.…

    • 1220 Words
    • 7 Pages
    Good Essays
  • Powerful Essays

    The initial velocity of an object does not have anything to do with its acceleration due to gravity.…

    • 708 Words
    • 3 Pages
    Powerful Essays
  • Satisfactory Essays

    Homework #2 Solutions

    • 1721 Words
    • 7 Pages

    (a) Constant speed → a = 0, then equation 7 reads: N = M (0 + g) = M g = 686 N upward (b) a = m/s2 upward. Equation 7 now reads: N = M 2.0 m/s2 + 9.80 m/s2 = 826 N upward (c) a = m/s2 downward. Equation 7 now reads: N = M −2.0 m/s2 + 9.80 m/s2 = 546 N upward (d) Free fall → a = −g: N = M (−g + g) = 0…

    • 1721 Words
    • 7 Pages
    Satisfactory Essays
  • Good Essays

    a. Suppose the particle moves so that the x-component of its velocity has the constant value vx = C; that is, x = Ct…

    • 614 Words
    • 3 Pages
    Good Essays
  • Good Essays

    Throwing a Football

    • 666 Words
    • 3 Pages

    When the football travels through the air for a long pass it always follows a curved path because the force of gravity influences the movement of the ball in the vertical direction. As the ball travels up, gravity slows it down until it stops briefly at its peak height; the ball then comes down, and gravity accelerates it until it hits the ground. Projectile motion is the path of any object that is launched or thrown and has an arched course (howstuffworks)…

    • 666 Words
    • 3 Pages
    Good Essays
  • Powerful Essays

    The dependence of two-dimensional velocity and position components on time as well as whether or not said components are constant was determined. Video was recorded of a ball being projected laterally and data of its flight were plotted. From analyzing the data and the corresponding graphs that followed, it was shown that while the horizontal and vertical components of position, and the vertical component of velocity have slope and therefore, are not constant, the graph of horizontal velocity showed no slope. From this it was concluded that horizontal velocity of a projectile remains constant.…

    • 1631 Words
    • 5 Pages
    Powerful Essays
  • Better Essays

    The Ballistic Pendulum

    • 1274 Words
    • 6 Pages

    The projectile with initial horizontal velocity v₀ is projected in such a way that its vertical distance of fall is H and the range is R. The trajectory is illustrated in the diagram below.…

    • 1274 Words
    • 6 Pages
    Better Essays