Physics of Projectile in Motion

A projectile is an object propelled by the application of an external force, which is allowed to move freely under the influence of gravity (and air resistance) . The object is launched into the air with a speed \( v_0 \) and at an angle \(\theta\) (as measured from a horizontal x-axis). The velocity vector \(\vec{ v_0 } \) can be resolved into two components $$ \text{Horizontal components, } v_x = v_0 \cos\theta $$ $$ \text{Vertical components, } v_y = v_0 \sin\theta$$ During flight, the projectile is acted upon by the acceleration due to gravity \(g\) alone. Hence, $$ \text{Horizontal components, } a_x = 0$$ $$ \text{Vertical components, } a_y = g$$ resulting in a changing vertical velocity \( v_y \) value where as the horizontal components \( v_x \) remains the same.

Projectile Equations

The trajectory of the projectile is given by the eqn., $$ y = y_0 + (\tan \theta )x - {gx^2 \over 2(v_0 \cos \theta)^2}$$ The trajectory of the projectile is given by the eqn., $$ y = y_0 + (\tan \theta )x - {gx^2 \over 2(v_0 \cos\theta)^2}$$ where \(g\), \(y_0\), \(\theta\), and \( v_0 \) are constants.

Maximum vertical height \(H\) is given by the eqn. $$ H = y_0 + {(v_0 \sin \theta)^2 \over 2g}$$

The time of flight \(T\) for the projectile is given by the eqn., $$ T = 2 {v_0 \sin \theta \over g} $$ (or twice the time taken to achieve the maximum vertical height \(H\).)

When the launch height is \(y_0 = 0\), the maximum horizontal range \( R \) is given by the eqns $$R = (v_0 \cos θ) t$$ and $$0 = (v_0 \sin θ)t - {1 \over 2} gt^2$$ Equating the above two eqns. $$ R = {v_0^2 \over g}\sin 2θ $$ The maximum horizontal range \(R\) can also be calulated from the total time of flight \(T\) and horizontal component of initial velocity \(\vec{ v_0 } \) $$ R = {\text{speed} \times \text{time}} = {v_0 \cos \theta \times T }$$