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Introduction to Range Equation for Projectile Motion

Projectile motion can be best explained by the example of a ball thrown in air with an angle (theta) to the horizontal ground. The path which a ball travels when it is thrown obliquely in air is the best example of projectile motion. Similarly, all objects, when released freely under the influence of gravity or any other attractive force, will follow a path which is similar to that of a ball thrown obliquely in air. One notices that when you throw a ball obliquely in air, it travels in a curved path which is called a parabola in Geometry.

In the following diagram, the range of a projectile is represented by 'x' meters; its maximum height by 'h' meters, and its initial velocity by 'u' ms/.


  Projectile Path


Finding the Range Equation for Projectile Motion:


To derive the range equation for projectile motion, we use the concept of dividing the initial velocity of a projectile into two parts, the horizontal velocity and the vertical velocity. Since gravity acts on the projectile in the downward direction, therefore the only the vertical velocity of the projectile is affected by gravity, not the horizontal velocity. This is because a force acting perpendicular to a vector will not produce any change in that vector since the work done will be zero. Thus, the gravity acting perpendicular to the horizontal velocity will not be able to change it.


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Derive the Range Equation for Projectile Motion:


Since we now know that the horizontal velocity of a projectile is always constant, therefore we can directly apply the formula velocity = (displacement)/(time).

Let the time duration from the release of the projectile to the point that it touches ground level be 't' seconds, and its initial velocity be '`u` ' m/s.

The initial velocity u m/s is divided into two:

  • the vertical velocity given by `u sin (theta)`

  • the horizontal velocity given by `u cos (theta)` .

Thus, we obtain

`v = d/t`

`u cos (theta) = x/t`

`x = t * u cos (theta)`

Thus, the equation of range of the projectile motion, that is, the horizontal distance travelled by it, is given by

`x = t * u cos (theta)`


I am planning to write more post on Horizontal Projectile Motion, Projectile Motion Graph  Keep checking my blog.