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Analysis of projectile motion

There is a very useful aspect of two dimensional motion that can be used with great effect. Two dimensional motion can be resolved in to two linear motions in two mutually perpendicular directions : (i) one along horizontal direction and (ii) the other along vertical direction. The linear motion in each direction can, then, be analyzed with the help of equivalent scalar system, in which signs of the attributes of the motion represent direction.

We can analyze the projectile motion with the help of equations of motion. As the motion occurs in two dimensions, we need to use vector equations and interpret them either graphically or algebraically as per the vector rules. We know that algebraic methods consisting of component vectors render vectors analysis in relatively simpler way. Still, vector algebra requires certain level of skills to manipulate vector components in two directions.

In the nutshell, the study of projectile motion is equivalent to two independent linear motions. This paradigm simplifies the analysis of projectile motion a great deal. Moreover, this equivalent construct is not merely a mathematical construct, but is a physically verifiable fact. The motions in vertical and horizontal directions are indeed independent of each other.

To illustrate this, let us consider the flight of two identical balls, which are initiated in motion at the same time. One ball is dropped vertically and another is projected in horizontal direction with some finite velocity from the same height. It is found that both balls reach the ground at the same time and also their elevations above the ground are same at all times during the motion.

Projectile motion

Comparing vertical and projectile motion

The fact that two balls reach the ground simultaneously and that their elevations from the ground during the motion at all times are same, point to the important aspect of the motion that vertical motion in either of the two motions are identical. This implies that the horizontal motion of the second ball does not interfere with its vertical motion. By extension, we can also say that the vertical motion of the second ball does not interfere with its horizontal motion.

Projectile motion and equations of motion

Here, we describe the projectile motion with the help of a two dimensional rectangular coordinate system such that (This not not a requirement. One can choose reference coordinate system to one's convenience):

  • Origin of the coordinate system coincides with point of projection.
  • The x – axis represents horizontal direction.
  • The y – axis represents vertical direction.

Initial velocity

Let us consider that the projectile is thrown with a velocity “u” at an angle θ from the horizontal direction as shown in the figure. The component of initial velocity in the two directions are :

u x = u cos θ u y = u sin θ

Projectile motion

Component projection velocities in x and y directions

Equations of motion in vertical direction

Motion in vertical direction is moderated by the constant force due to gravity. This motion, therefore, is described by one dimensional equations of motion for constant acceleration.

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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