Horizontal projectile motion, Derivation

Horizontal Projectile Motion

(Content copied exactly from the board)

At t = 0 , object is at O

Velocity of projectile at any time t

(using 1st equation of motion)

\(v_x = u + a_x t\)

Here \(a_x = 0\), so

\(V_x = u \tag{1}\)

\(v_y = u_y + a_y t\)

\(V_y = 0 + g t = g t \tag{2}\)

Resultant velocity:\\ \(V = \sqrt{V_x^2 + V_y^2} = \sqrt{u^2 + g^2 t^2} \tag{3}\)

Displacement of Projectile & Trajectory

Using 2nd equation of motion:

\(x = ut \tag{1}\)

\(y = 0 \cdot t + \dfrac12 g t^2 = \dfrac12 g t^2 \tag{2}\)

From (1):

\(t = \dfrac{x}{u}\)

Substituting in (2):

\(y = \dfrac{g x^2}{2 u^2} \tag{4}\)

This is the equation of a parabola ⇒ Path of projectile is parabolic.

Time of Flight of Projectile

(using 2nd equation of motion)

\(s = u t + \dfrac12 a t^2\)

Here \(s = h,\; a = g,\; u = 0\):

\(h = 0 \cdot T + \dfrac12 g T^2\)

\(T^2 = \dfrac{2h}{g}\)

\(T = \sqrt{\dfrac{2h}{g}} \tag{5}\)

Range of Projectile

\(\text{Range} = \text{speed} \times \text{time of flight}\)

\(R = u \cdot T\)

\(R = u \sqrt{\dfrac{2h}{g}} \tag{6}\)

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