Question

Show that for a projection angle θ, where 0° < θ <90°, the path of the projectile is a parabola.

Answer 1

AnswEr :

We have to show that the Trajectory of a projectile is Parabolic

Let the initial velocity of the projectile be u and ang!e of projection be alpha .

Resolving the initial velocity vector along X and Y axes :

• Along X Axis : u cos∅

• Along Y axis : u sin∅

Acceleration along the axes :

• Along X axis : 0

• Along Y axis : - g

Displacement along x - axis :

$\sf \: x = u_xt + \dfrac{1}{2} {at}^{2} \\ \\ \longmapsto \: \sf \: x = (u cos \: \theta )t \\ \\ \longmapsto \: \sf \: t = \frac{x}{u cos \: \theta }$

Displacement along y - axis :

$\sf \: y = u_yt + \dfrac{1}{2} a {t}^{2} \\ \\ \longrightarrow \: \sf \: y = usin \: \theta \times \dfrac{x}{ucos \: \theta } - \frac{g {x}^{2} }{2 {u}^{2} cos {}^{2} \theta } \\ \\ \longrightarrow \: \sf \: y = x \: tan \: \theta - \dfrac{g {x}^{2} }{ {2u}^{2} {cos}^{2} \theta }$

Equation of a parabola is given as :

y = ax - bx²

Thus,

• $\sf \: a = tan \: \theta$
• $\sf \: b = - \dfrac{g}{2 {u}^{2} {cos}^{2} \theta }$

Thus,the Path of a Projectile is Parabolic