Parabolic movement

Parabolic movement . Movement made by an object whose trajectory describes a parabola .


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  • 1 General
  • 2 Types of parabolic movement
    • 1 Semi-parabolic movement
    • 2 Parabolic motion (complete)
  • 3 equations
    • 1 Equation of acceleration
    • 2 Equation of velocity
    • 3 Equation of position
  • 4 Parabolic motion with friction
  • 5 Sources


The movement Parabolic corresponds to the ideal trajectory of a projectile moving in a medium that offers no resistance to the advance and is subject to a uniform gravitational field.

It can be analyzed as the composition of two rectilinear movements: a horizontal uniform rectilinear movement and a uniformly accelerated vertical rectilinear movement .

Types of parabolic movement

Semi-parabolic movement

The movement of parabola or semiparabólico (horizontal pitch) can be considered as the composition of a uniform rectilinear horizontal advance and the free fall of a body at rest.

Semi-parabolic movement.

Parabolic motion (complete)

The movement full parabolic can be considered as the composition of a uniform rectilinear horizontal advance and a vertical pitch up, a uniformly accelerated rectilinear motion downwards (MRUA) by the action of gravity .

Under ideal conditions of resistance to zero advance and uniform gravitational field, the above implies that:

  1. A body that falls freely and another that is thrown horizontally from the same height take the same time to reach the ground.
  2. The independence of the massin free fall and vertical launch is just as valid in parabolic movements.
  3. A body thrown vertically upwards and another parabolically complete that reaches the same height takes the same to fall.


There are two equations that govern parabolic motion :

Parabolic motion equations.

Graphical representation of the equations.

The initial velocity can be expressed in this way:

Equation 1.

It will be the one used, except in cases where the angle of the initial speed must be taken into account.

Acceleration equation

The only acceleration involved in this movement is that of gravity , which corresponds to the equation:

Acceleration equation.

, which is vertical and down.

Velocity equation

The speed of a body following a parabolic path can be obtained by integrating the following equation:

Equation of speed.

The integration is very simple because it is a first order differential equation and the final result is:

Result of the velocity equation.

We start from the value of the acceleration of gravity and the definition of acceleration

and we have

Derivation of the velocity equation.

This equation determines the speed of the mobile as a function of time , the horizontal component does not vary, while the vertical component does depend on time and the acceleration of gravity .

Equation of position

Starting from the equation that establishes the speed of the mobile with the relation to time and the definition of speed, the position can be found by integrating the following differential equation:

Equation of position.

The integration is very simple because it is a first order differential equation and the final result is:

Result of the equation.

The path of the parabolic movement is formed by the combination of two movements, one horizontal of constant speed, and the other vertically uniformly accelerated; the conjugation of the two results in a parabola.

Parabolic motion with friction

When we consider friction the trajectory is almost a parabola but not exactly. The study of the trajectory in that case is considered by the ballistics.


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