Projectile motion  (Page 5/6)

Equations of motion in horizontal direction

The force due to gravity has no component in horizontal direction. Since gravity is the only force acting on the projectile, this means that the motion in horizontal direction is not accelerated. Therefore, the motion in horizontal direction is an uniform motion. This implies that the component of velocity in x-direction is constant. As such, the position or displacement in x-direction at a given time “t” is :

This equation gives the value of horizontal position or displacement at any given instant.

Displacement of projectile

The displacement of projectile is obtained by vector addition of displacements in x and y direction. The magnitude of displacement of the projectile from the origin at any given instant is :

The angle that displacement vector subtends on x-axis is :

Velocity of projectile

The velocity of projectile is obtained by vector addition of velocities in x and y direction. Since component velocities are mutually perpendicular to each other, we can find magnitude of velocity of the projectile at any given instant, applying Pythagoras theorem :

The angle that the resultant velocity subtends on x-axis is :

Equation of the path of projectile

Equation of projectile path is a relationship between “x” and “y”. The x and y – coordinates are given by equations,

$$\begin{array}{l}y=u_{y}t-\frac{1}{2}g{t}^2\\ x=u_{x}t\end{array}$$

Eliminating “t” from two equations, we have :

For a given initial velocity and angle of projection, the equation reduces to the form of $y=Ax+B{x}^2$ , where A and B are constants. The equation of “y” in “x” is the equation of parabola. Hence, path of the projectile motion is a parabola. Also, putting expressions for initial velocity components $u_x=u\cos \theta \mathrm{and}{u}_y=u\sin \theta$ , we have :

Some other forms of the equation of projectile are :

Exercises